A Projection Method for Two-Phase Incompressible Flow with...

24
Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum ur Informationstechnik Berlin MICHAEL OEVERMANN ,RUPERT KLEIN MARSHA B ERGER ,J ONATHAN GOODMAN A Projection Method for Two-Phase Incompressible Flow with Surface Tension and Sharp Interface Resolution Konrad-Zuse Zentrum Courant Instituteof Mathematical Sciences, 251 Mercer Street, New York, NY 10027, U.S.A. ZIB-Report 00–17 (May 2000)

Transcript of A Projection Method for Two-Phase Incompressible Flow with...

Page 1: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

Takustraße 7D-14195 Berlin-Dahlem

GermanyKonrad-Zuse-Zentrumfur Informationstechnik Berlin

MICHAEL OEVERMANN�

, RUPERT KLEIN�

MARSHA BERGER�

, JONATHAN GOODMAN�

A Projection Method for Two-PhaseIncompressible Flow with Surface Tension and

Sharp Interface Resolution

�Konrad-Zuse Zentrum�Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10027, U.S.A.

ZIB-Report 00–17 (May 2000)

Page 2: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

� ��������� ���������������������������! "�$#���&%(')�+*,��-�/.10����-'�')��2�3��5463��7 8�9�:��;=<���>%?��@�A���B'������C%(�B�D;=�E%?��0D*,�F������>%?��HG"��')�/3�<I�:�����

JLKNMPORQTSVU?W&SVXTS>Y[Z6Q]\^\`_�aIb^c�SVYedgfEUhSVKN\fji]\^YeQTk�lnm-b^oeSpm�S>\�deY[b^Zq_Rr�Qts)b^o[deY,u=v)_�wyx:w{zT|~}�S>YeU�K�\`_ �jSVYeZ6Q]\��

J�Q]YPonORQ6}=S>Ye�TS>Y,_R�i]\RQ]d[ORQ]\��ji)i�k�Z�Qt\� i]b:YPQ]\�dI��\Ro[d[KNd[b^dPSt_-�T|)w�J�SVYPM,SVYE��deYPS>S>d,_ �BS>�@�/i]Yes _R�g�1w{�]��Tv�_���u��-u��pu

J�Qy�L�]�:_R�]�T�T�

�/��9���n�,�e� �¡/¢>£¤¡9¥¦¡�§�¨(©Iª�«,¬�¬®­&¥¦¡9¯�°[§y±j°[£²±V¡�£?¢>£¤°�³´¡9¯�¨¤µ¶°[§�·?¡�¨¦¸¹°�±&ªº°n£(¨¦¸¹¡/¥¦µ¶·»«¹¬¶©[¨¤µ¶°[§°nªR¨h¼:°[½¾¢,¸,©e¥¦¡�µ¶§,¯9°[·$¢,£¤¡9¥¦¥¦µ®¿¹¬¶¡?À¹°�¼q¼�µ¶¨¦¸E¥´«>£¦ªº©[¯9¡/¨¤¡Á§¹¥¦µ¶°[§ÃÂIÄ:¸¹¡�Å�©�Æ�µ¶¡�£¦½hÇ�¨¤°[ÈP¡9¥¡9ÉÊ«,©[¨¤µ¶°[§,¥�©n£¤¡�¥¦°e¬¶ÆP¡�±&¼�µ®¨¦¸j©g¢>£¤°�³´¡9¯�¨¤µ¶°[§�·(¡�¨¦¸,°�±j°n§�©gË>Ì>¡9±jÍ:©[£¦¨¤¡9¥¦µ¶©[§pÎ[£¤µ�±yÂÄ:¸,¡:ª�£¤¡�¡^µ¶§�¨¤¡�£¦ª�©e¯9¡�¿{¡�¨h¼:¡9¡�§»¨¦¸¹¡:¨N¼^°-À>«¹µ�±>¥tµ�¥]¨¦£¤©[¯²ÈP¡�±7¼�µ®¨¦¸�©�¬¶¡9ÆP¡9¬Ê¥¦¡�¨�©n¢,¢>£¤°e©e¯¤¸ÃÂÄ:¸,¡7¯9°[§¹±>µ®¨¤µ�°n§¹¥�©[¨ ¨¦¸¹¡`µ®§�¨¤¡�£¦ªº©[¯9¡`ªº°[£-¨¦¸¹¡Ï¢>£¤¡9¥¦¥´«,£¤¡nÐy¨¦¸¹¡`¢,£¤¡9¥¦¥´«>£¤¡7Î[£¤©P±Vµ�¡Á§V¨�Ðy©[§¹±¨¦¸,¡$ÆP¡�¬�°Ê¯9µ®¨h­�©[£¤¡?¡ÁÌV¢,¬�µ¶¯9µ®¨¤¬¶­Fµ®§¹¯9°n£¦¢{°[£¤©[¨¤¡9±gµ®§�¨¤°/¨¦¸,¡$¥¦¯¤¸¹¡�·(¡$¬�¡9©e±>µ®§¹Î(¨¤°�©/¥´¸¹©n£¦¢£¤¡Á¢,£¤¡9¥¦¡�§�¨¤©n¨¤µ�°n§&°[ªR¨¦¸,¡?¢,£¤¡9¥¦¥´«>£¤¡�±>µ¶¥¦¯9°[§�¨¤µ®§�«¹µ®¨h­�©n§y±g¨¦¸,¡(µ¶§�¨¤¡�£¦ª�©e¯9µ¶©e¬�ªº°n£¤¯9¡nÂIÄ:¸¹¡

¥¦¯¤¸¹¡Á·(¡Fµ®§&¨¦¸,¡�¢>£¤¡9¥¦¡�§�¨¤¡�±�ª�°[£¦·Ñ±>°Ê¡9¥»§¹°[¨$µ¶§�¨¦£¤°�±�«¹¯9¡(©e±,±>µ®¨¤µ¶°[§¹©[¬�¢{°[µ¶§�¨¤¥7µ¶§E¨¦¸,¡¥´¨¤©n§y±V©[£²±�Ë>§¹µ®¨¤¡^±>µ®Ò{¡�£¤¡�§,¯9¡^¥´¨¤¡�§,¯9µ¶¬�¥ÁÂ)Í:°[·?¢>«,¨¤©[¨¤µ¶°[§,©e¬V£¤¡9¥´«¹¬®¨¤¥©[£¤¡�¯9°n·?¢¹©n£¤¡�±�¼�µ®¨¦¸©n§¹©[¬¶­�¨¤µ¶¯=¥¦°e¬®«,¨¤µ¶°[§,¥?ªº°[£�©E¥´¨¤©[¨¤µ¶¯=£¤°[«>§y±p¿>«,¿,¿,¬�¡�Ð ±>©n·?¢{¡�±�¥´«>£¦ªº©[¯9¡g¼:©�ÆP¡9¥ÁÐ�©[§¹±Ó ©�­V¬�¡9µ¶Î[¸>½NÄ]©�­>¬¶°[£:µ¶§,¥´¨¤©[¿,µ�¬¶µ¶¨¤µ¶¡9¥ÁÂ

Ô Õ�Ö=×�Ø�ÙgÚEÛEÜ�×:ÝnÙFÖÞ¤ßpà�á{âäã?åÃæ�å]çnè/é-çIêyç[ëVç[ì�í>åpæ&î>ì�í>ïÃæ�ìºìºð&ã²çeñ�íVßtêpí>è�ê{çnè?ß¹ò{ó�çnè�âäñnæ�ì^ó�çnàÁá{íyêpô¾íVè?à¤é�íEå{áÃæ>ã²ç=âºßÃñ�í>ó=õå{èÁçeãÁãÁâºï{ìºçRöÃíÊé7ã:é`â�àÁá=ã²òÃè²ô�æ>ñ�ç-àÁçnßtã²âºíVß�æ�ßÃêFá{âºîVá�êyçnßtã²â�à¤ð�æ�ßtê�ë¹âäãÁñní>ãÁâ�à¤ð$è9æÊàÁâºí,ã�÷^ø�òÃè:î>í>æVì,âäã�àÁí?áÃæPë>çæ�ó�çnàÁá{íyê/àÁáÃæ�à�âäã^á{âºîVá/í>è�ê{çnèRé`á{âºìºç-óIæVâºß>à9æ�âºß{âºß{î7ãÁáÃæ�è�å�â�ß,à�çnèÁô�æVñ�çeã�÷�ù è�çnë¹âºíVòÃã�ó�ç�à�á{íyê{ã^çnâ�àÁáÃçnèRêyâäêß{í�à»æ>ñ9á{âºçnëVç$ã²çeñ�í>ßÃêgí>è�ê{çnè7æVñ[ñ�ò{è9æVñnð�íVè7êyíIãÁí�æÊà`à�á{ç�ñ�í,ã¤à7íVô:ãÁå{èÁçeæVêyâºß{îFàÁáÃç(â�ß,à�çnèÁô�æVñ�ç?í>è7ã²ò{èÁô�æVñnçàÁçnßtã²âºíVß/ô¾íVè9ñ�ç�íÊëVç[è�óIæVß,ð(î>èÁâäê�ñnçnìºìäã�÷)ú�ç7ãÁíVìºëVç àÁá{çÏß{í>ßyõ´ì�âºß{çeæ�è�û�æPë¹âºçnèÁõ¦ü,à�íVý>ç[ã�çeþ,òÃæÊà�â�í>ßÃã�íVß=æ�òÃß{â�õô¾íVè�ó ÿ�æVè²à�ç[ãÁâäæ�ß=îVè�âäê=é`â�à�ágæFå{èÁí � ç[ñ�àÁâºíVß=ó�ç�à�á{íyêT÷��Ïá{ç�à¤é-í/öÃò{âäê�å{áÃæ>ã²çeã-æVèÁç$ñ�í>ò{å{ìºç[ê=ï¹ð=ç��yå{ìºâäñ�â�àâ�ßtñ�íVè�å]íVè9æÊàÁâºí>ß/í�ôà�á{ç$ý,ßÃíÊé`ß � ò{ó�ågñníVßÃêyâ�à�â�í>ßÃã:ô¾íVè-àÁáÃç�å{è�ç[ã�ãÁò{èÁç»â�ß,à�í(àÁá{ç?êyâ��Tçnè�çnßÃñnç»ãÁñ9áÃçnó�ç�÷�ú�çåtç[è²ô¾í>èÁó æ�ß�æVãÁð¹ó�åyàÁíVàÁâäñ7æ�ßtæ�ìºðyã²âäãRíVô)àÁáÃç�çeþ>òtæÊàÁâºí>ßÃã âºß�à�á{ç?ìºâ�ó�â�à�íVô�æ�ßBâ�ß�Ãß{â�àÁç$êyçnßÃãÁâ�à¤ð�è�æ�àÁâºí/æVßÃêòÃã²ç�àÁáÃç$è�ç[ãÁò{ì�à�ã-í�ô�àÁá{âäãÏæ�ßtæ�ìºðyã²âäã�æ>ã�æ�î>ò{âäêyç7ô¾í>è-à�á{ç(ñ�í>ßÃã¤à�èÁòtñ�àÁâºí>ßIí�ô:íVò{èÏó�ç�à�á{íyêT÷�7ì�àÁáÃíVò{î>ágíVòÃèó�í�àÁâºëÊæÊà�âºíVßEæ�ßÃêpíVòÃè?æ>ã²ð¹ó�åyà�í�à�âºñ[ã¤õ´ïÃæ>ã²çeê�æVå{å{è�í>æ>ñ9áEàÁí�àÁáÃç�å{èÁí>ï{ìºçnó æ�è�ç=êyâ��]ç[èÁç[ß>à$ô¾è�íVó��jæPð>í ã� ����� í>è�à�áÃæÊà�í�ô��)ç��Rç[þ,ò{ç$æ�ßÃê���â ���������� ��!� à�á{ç»è�ç[ãÁò{ì�àÁâºß{î(ß¹ò{ó�ç[èÁâäñnæVì{ãÁñ9áÃçnó�ç7áÃæ>ã ãÁíVó�ç7ãÁâºó�â�ìäæVèÁâ�àÁâºçeãâ�ß/àÁá{ç`éÏæPð(â�àRâ�ßtñ�íVè�å]íVè9æÊàÁçeã � òÃó�åÃã^í�ô]å{èÁçeãÁãÁò{è�ç�æVßÃêFå{èÁçeãÁãÁò{è�çÏîVè9æVêyâºçnß,à9ã�÷Þ¤ó�å]íVèÁà�æVß>à�ìºð � é-çÏç[ó�å{ìºíÊðàÁá{ç(ß{íVõ´ãÁìºâºågñ�í>ßÃêyâ�àÁâºíVß�æÊà`à�á{ç(âºß>à�çnèÁô�æVñnç?æVßÃêgàÁá{ç�è�ç[ãÁò{ì�àÁâºß{î � ò{ó�åEñ�í>ßÃêyâ�àÁâºíVßgô¾íVè`ï]í�à�áBà�á{ç�ß{í>èÁóIæVìæ�ßÃêIàÁá{ç$à�æVß{îVç[ß,àÁâäæ�ì]å{è�ç[ã�ã²ò{è�ç»î>è�æ>êyâºçnß,à�ã�àÁí=è�ç[êyòÃñnç»à�á{ç(ñ�í>ó�å{ìºç��yâ�à¤ð/í�ô�àÁá{ç"�:æ�å{ìäæVñnç�ã²àÁç[ßÃñ�âºì]â�ßgàÁá{çå{èÁí � ç[ñ�àÁâºíVß&ã¤à�çnå÷#�Ïá{ç/ñ�òÃèÁè�çnß,à»ë>çnè9ã²âºíVßBí�ôRà�á{çFã�ñ9á{çnó�ç�æ>ñ9á{âºçnë>ç[ã�ìºí¹ñ[æ�ì$�tè�ã²à � îVìºíVïtæ�ì�ãÁç[ñ�í>ßÃê&íVè9êyç[èíVßEà�á{çFïtæVãÁâºã»í�ô-æBã¤à9æ�ßÃêÃæ�è9ê¹õ!�tëVç�õ´å]íVâºß,à»ã¤à�çnßÃñnâ�ì�íVß�æ�ÿ�æ�èÁàÁçeã²âäæVß�ó�çeã²á�÷"�Ïá{ç�ó�íÊë¹âºß{î�âºß>à�çnèÁô�æVñnç�âäãàÁè9æVñ9ýVçeê�ë¹âäæ&æ&ìºçnëVç[ì�ã²çnà/ó�ç�à�á{íyê � æVßÃê�ô¾íVè�àÁá{çgãÁíVìºòyà�âºíVß�íVôÏàÁá{çIåÃèÁçeãÁãÁò{è�ç�ùRí>âäãÁãÁíVßjçeþ,òÃæÊà�â�í>ßÃã?é�çæ�å{å{ìºðIæ�ó�ò{ì�àÁâ�õhî>èÁâäêFàÁçeñ9á{ß{âäþ,ò{ç�÷

Page 3: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

�Ïé�í�å{áÃæVãÁç=âºßÃñ�í>ó�å{èÁçeãÁãÁâºï{ìºç$öÃíÊé å{ìäæPðyã$æ�ß�â�ó�å]íVèÁà�æVß,à�è�íVìºç/âºß�óIæVß¹ð&àÁç[ñ9áÃß{âäñnæ�ìRæVå{å{ìºâäñnæÊà�âºíVßÃã� ìºâºýVç�âºß{ý,õ � ç�à�åÃèÁâºß,àÁâºß{î � �Ãï]çnè(åÃèÁíyêyòÃñ�àÁâºíVß � ó�â��yâºß{î�å{è�íyñ�ç[ã�ãÁç[ã � ãÁå{è9æPðyã � ô¾ò{çnì�ñníVó�ï{òÃã²àÁâºíVß � çnà�ñÊ÷�� æVãé-ç[ì�ì^æVã$âºß�óIæVß¹ð�ïÃè�æVßÃñ9á{ç[ã$íVô àÁáÃç�ßtæÊàÁòÃè�æVì^ã�ñ�âºç[ßÃñ�çeã � ç�÷ î{÷ � êyð¹ßÃæ�ó�âäñ[ã7í�ôÏêyè�íVå{ìºç�à9ã � âºßÃã²à�æVï{âºì�â�à¤ðBí�ôã¤à�è�æ�àÁâ��Ãç[ê�ötíÊé7ã�âºßEà�á{ç=æÊàÁó�í,ã²åÃá{çnè�ç � æVñ[ñ�ò{ó�ò{ìäæÊà�â�í>ßBíVô å]íVìºìºòyà�æVß>à9ã»âºßjè9æ�âºßjêyè�íVåtã���÷ �Ïá{ç�ñníVó�å{ò{õà�æÊà�âºíVß íVô»à¤é�í�õ´å{áÃæVãÁç�â�ßtñ�íVó�åÃèÁçeãÁãÁâºï{ìºçFötíÊé é`â�àÁá ìºæVèÁî>çgêyç[ßÃã²â�à¤ð~è9æÊà�â�í�æ�ßÃê ãÁò{è²ô�æ>ñ�çBàÁç[ßÃã²âºí>ß6âäã=æñ9áÃæ�ìºìºçnß{î>âºß{îgå{è�íVïÃì�ç[ó+æVßÃê~áÃæ>ã/æÊàÁàÁè9æVñ�àÁç[ê óIæVß¹ð�è�ç[ãÁç[æ�è9ñ9á{ç[è�ã?íÊë>çnè(à�á{çgìäæVã²à/ð>ç[æ�è9ã � ãÁçnç � � � � æVßÃê� � ��� ô¾íVè`æ�ß�íÊëVç[èÁë¹âºçné æVßÃêBê{ç�à�æVâºì�çeê�è�ç�ô¾çnè�çnßtñ�ç$ì�âäã²à�ãn÷�û�ò{ó�çnè�âäñnæ�ì]ó�ç�à�á{íyê{ã-ô¾íVèÏà¤é�í�õ´å{áÃæ>ã²ç»öÃíÊéqñnæVßïtç�êyâºë¹âäêyç[êBï{è�í>æ>êyìºð�âºß,àÁí�à¤é�íIñ�ìäæVã�ãÁç[ãn÷:Þ¤ß ��yçeê¹õhî>èÁâäêgó�ç�àÁáÃí¹êÃã�àÁá{ç�çeþ,òÃæÊà�â�í>ßÃãÏæ�è�ç?òÃã²òtæ�ìºìºðIã²í>ì�ë>ç[êâ�ß�à�á{çgé`á{í>ìºçIêyíVóIæVâ�ß�íVß æjå{èÁçeêyç��tß{ç[ê ��yçeê~îVè�âäêT÷ �ÏáÃçgó�í>ã²à�å]íVå{ò{ìäæVè�ó�ç�à�á{íyê{ã�íVô�à�á{âäã(à¤ð¹å]çâ�ßtñ�ìºòÃêyç=à�á{çIëVí>ìºò{ó�ç�õ´í�ôäõhöÃò{âäê{ã � ��ø�����ó�ç�àÁáÃí¹êÃã � � ���� ���� � ì�ç[ëVç[ì ã²çnà/ó�ç�à�á{íyê{ã � ����� �����!� ô¾è�íVß,àÁõàÁè9æVñ9ý¹âºß{îgã�ñ9á{çnó�çeã � �� ��� � � ����� � æ�ßtêjà�á{ç ��æ�è�ýVçnèÁõ¦æ�ßÃê¹õ²ÿ-çnìºì�ó�çnàÁá{íyê � ��� � ÷ �Ïá{ç=ã²çeñ�íVßtê�ñ�ìäæ>ãÁã$æ�è�çàÁá{ç�ó�íÊë,âºß{îVõhî>èÁâäêFæ�ßÃê��:æ�î>è�æVß{îVâäæ�ß�ó�ç�à�á{íyê{ã�é`áÃçnè�ç�æ/ãÁçnåtæ�è9æÊàÁç � ï]íVòÃßÃê{æ�è�ð �{à²à�ç[êgîVè�âºê=âäã òÃã²çeê=ô¾íVèç[æVñ9á=å{átæVãÁç � ��� � � ��� ÷ »ì�àÁá{í>ò{îVá�ó�íÊë¹âºß{î�õ´îVè�âäê$àÁçeñ9á{ß{âäþ,ò{ç[ãRí��]ç[è å]í�à�çnß,àÁâäæ�ìºìºð?àÁáÃç7á{âºîVá{çeã¤à æ>ñnñnò{è�æ>ñ�ð �àÁá{ç[ðpæ�è�ç=êyâ���ñnò{ì�à?àÁíEæ�å{åÃì�ðjô¾íVè�ñ�í>ó�å{ìºç��Eô¾ò{ìºìºð&àÁáÃèÁç[ç�õ¦êyâºó�çnßÃãÁâºíVßÃæVì�å{è�íVï{ìºçnóIã�íVè(åÃèÁí>ï{ìºçnóIã�é`â�à�áàÁíVå]íVìºí>îVð�ñ9áÃæ�ßÃîVç[ãn÷

ø�ßÃç7í�ô]àÁá{ç»óIæ � íVè�ê{â��Iñnò{ì�àÁâºç[ã:âºß�ñ�íVó�åÃòyàÁâºß{î$à¤é�í�õ´å{áÃæVãÁç�öÃíÊé7ãRâäã:àÁáÃç�ãÁâ�ßÃîVò{ìäæ�è^ãÁò{è²ô�æ>ñ�çÏà�çnßÃãÁâºíVßàÁçnè�ó@âºßpàÁá{çgû�æPë¹âºçnèÁõ´ã²àÁí>ýVç[ã»ç[þ,òÃæ�àÁâºíVßÃãn÷�Þ¤ß óIæ�ß¹ðEß¹ò{ó�çnè�âäñnæVì:ó�çnàÁá{íyê{ã?í>ß ��yç[ê�îVè�âäê{ã � ç�÷ î{÷ � �� ���� � � � ��� � ��������� ����� �!� àÁá{çIãÁò{èÁô�æVñnç/à�çnßÃãÁâºíVß�âºã$à9æ�ý>çnß�â�ß,à�í�æ>ñnñ�í>ò{ß,à(é`â�à�ápß¹ò{ó�ç[èÁí>òÃã?êyçnè�âºëÊæÊàÁâºë>ç[ãí�ô/ùRçeã²ý¹âºß$ ãIâºó�ó�çnè9ã²çeê6ï]íVòÃßÃê{æ�è�ð ó�ç�à�á{íyê � � ��� ÷ �Ïá{âäãBó�ç�àÁáÃí¹êqé�æ>ãgí>èÁâºî>â�ßtæ�ìºìºð ê{çnëVç[ìºíVå]ç[êAà�íó�í¹ê{çnì�ï{ìºí¹íyêBöÃíÊéLâºß�àÁáÃç�á{ç[æVè²à � é`á{ç[èÁç?à�á{ç�ç[ìºæ>ã¤à�âäñ�ï]íVòÃßÃê{æ�è�ðgç �yçnèÁà�ãÏô¾í>è�ñnç(íVß&àÁáÃç(öÃò{âäêT÷�Þ¤ß&àÁá{çâ�ó�ó�ç[è�ãÁç[êIïtí>ò{ßÃê{æVèÁðBó�ç�àÁáÃí¹ê � ã²âºß{î>ò{ìäæ�è�ô¾íVè9ñ�çeã7æ�è�ç$àÁè9æ�ßÃã²ô¾çnè�è�ç[êBàÁí=à�á{ç ��yç[ê&î>èÁâäêBòÃãÁâ�ßÃî=ê{âºã�ñ�è�ç�à�çêyçnì�à�æ�ô¾ò{ßtñ�àÁâºí>ßÃã�÷ �ÏðFòtã²âºß{î$àÁáÃç»êyâäãÁñnèÁçnàÁâ��nçeêFê{âºã�ñ�è�ç�à�ç`êyçnì�à�æ$ô¾òÃßÃñ�à�â�í>ßÃã � àÁáÃç»êyâäãÁñníVß,àÁâºß¹ò{â�à¤ð?âäã�ã²åÃèÁçeæVêíÊëVçnè:ã²ç[ëVç[è�æVìVîVè�âäê(ñ�ç[ì�ìäãn÷�ü¹âºß{îVòÃìºæVèô¾íVè9ñ�çeã�æVå{å]ç[æ�è:âºß(àÁáÃâºã�ó�ç�à�á{íyê/æ>ã)ë>íVìºò{ó�ç^ô¾íVè9ñ�çeã�÷�Þ¤ß�à�á{ç�ñníVß,àÁç �¹àí�ôTå{èÁí>ï{ìºçnóIã�âºß,ë>íVìºë¹âºß{î7ã²òÃè²ô�æ>ñ�ç�àÁç[ßÃã²âºí>ß/à�á{âäãRæ�å{å{è�í>æ>ñ9á/âäã^òÃãÁòÃæ�ìºìºð�ñnæVìºì�çeê�ñ�í>ß,àÁâºß¹ò{íVòÃã^ãÁò{èÁô�æVñ�ç�ô¾í>è�ñnç� ÿÏü�����ó�ç�à�á{íyê � é`á{âäñ9á�áÃæVãÏï]çnç[ß&â�ß,à�èÁíyêyòÃñnç[êBï¹ð��Ïè�æ>ñ9ý,ïÃâ�ìºì]ç�à»æVìN÷Râºß � � ÷ ��è�ð,î>îVëÊæVãÁíVßgç�à»æ�ìh÷ � �����æ�ßÃê$ùRíVåÃâ�ßÃç�àTç�à�æVìN÷ � ����� âºßÃñ�í>èÁå]íVè9æÊà�ç)à�á{ç ãÁò{èÁô�æVñ�ç:à�çnßÃãÁâºíVß�ô¾íVè9ñ�çRâºß(æ �Ãß{â�àÁçRë>íVìºò{ó�ç�ãÁç�àÁàÁâºß{îÃ÷�Þ¤ß?àÁá{ç[âºèæ�å{å{è�í>æ>ñ9áIà�á{ç/ãÁò{è²ô�æ>ñ�ç?à�çnßÃãÁâºíVßBô¾íVè9ñ�ç(âäã`ß{íVà»ã²ó�çeæ�è�ç[êBï{òyà»âäã`æ�åÃå{ìºâ�çeêBãÁíVìºç[ì�ð=à�í=àÁá{ç �Ãß{â�àÁç(ë>íVìºò{ó�çñ�çnìºìÏñ�í>ß>à9æ�âºß{âºß{î&à�á{çBâºß,àÁç[è²ô�æ>ñ�çV÷�Þ¦à�æ�åÃåtçeæ�è9ãFæ>ã�à�á{ç&å{èÁíyêyòtñ�à=í�ô`àÁáÃç&ãÁò{èÁô�æVñnçIàÁç[ßÃãÁâ�í>ß6ñ�í¹ç���ñnâºçnß,ààÁâºó�ç[ã àÁá{ç�ê{â �Tçnè�çnßtñ�ç?ïtçnà¤é-ç[çnß�àÁáÃç$âºß,àÁç[è²ô�æ>ñ�ç�à�æVß{îVç[ß>à9ã`æÊà`à�á{ç(ïtí>ò{ßÃê{æVèÁâºçeã íVô^æ=ñ�í>ß,àÁè�íVìTëVí>ì�òÃó�ç�÷Þ¤ß�àÁáÃç�ñníVß,àÁç �,à=í�ô$ã²í>ì�ë¹âºß{îEùRí>âºã�ãÁíVß$ ã�æ�ßtê à�á{çBïÃâ�átæ�è�ó�íVß{âäñ=ç[þ,òÃæ�àÁâºíVß~í>ß6âºè�èÁç[îVò{ìäæ�è(è�çnî>âºíVßÃã �

�jæPðVíIâºß � � � � ãÁá{íÊé�ç[êjægé�æPð�à�ígç��yå{ìºâäñ�â�à�ì�ðgâ�ßtñ�íVè�å]íVè9æÊàÁç(ý¹ß{íÊé`ß � ò{ó�åpñníVßÃêyâ�à�â�í>ßÃã`íÊë>çnè?êyâäãÁñníVß,àÁâ�õß,òÃâ�à�âºç[ã àÁí=îVçnàÏáÃâ�î>á{çnèÏí>è�êyç[è �Ãß{â�àÁç(êyâ��Tçnè�çnßÃñnç»ô¾í>èÁó�ò{ìäæVã ô¾íVè�àÁáÃç ��æ�åÃìºæ>ñ�ç$íVå]çnè9æÊà�íVè[÷ ��ç �Rç[þ,ò{ç(æVßÃê�)â âºß ������� å{èÁçeã²ç[ß,àÁç[ê~æEãÁç[ñníVßÃê�í>è�êyç[è�ó�ç�à�á{íyê�ô¾íVè�ç[ìºì�âºåyà�âäñ/çeþ>òtæÊàÁâºí>ßÃã?é`â�àÁá~êyâäãÁñníVß,àÁâºß¹ò{â�àÁâºçeã�æVßÃêã²âºß{î>ò{ìäæ�è-ãÁíVòÃè�ñnç[ãÏæ�ßÃê � â�ß�ã²åÃâ�è�â�à�íVô�ùRçeã²ý¹âºß$ ã ó�çnàÁá{íyê � ñ[æ�ìºìºç[ê�â�àÏàÁá{ç?âºó�ó�çnè9ãÁç[ê�âºß,àÁç[è²ô�æ>ñ�ç»ó�çnàÁá{íyêT÷�Ïá{çnâºè?ó�ç�à�á{íyê�âäã?ëVç[èÁðpã²âºó�âºìäæ�è`à�í �jæPðVí ã?æ�åÃå{èÁí,æVñ9á÷ �Ïá{çeã²ç=ó�çnàÁá{íyê{ã?í��]ç[è(àÁá{ç=ô¾è9æ�ó�ç[é-í>èÁý�à�íñ�íVßtã¤à�èÁòÃñ�à�á{âºî>á{çnè�íVè9êyçnè$ó�ç�à�á{íyê{ã7ô¾í>è»ötíÊé7ã»é`â�àÁápâºó�ó�çnè9ã²çeêgï]íVò{ßtê{æ�è�âºç[ã»æVßÃê�ê{âºã�ñ�í>ß,àÁâºß¹ò{íVòÃã»ã²íVõì�ò{àÁâºíVßÃã�ï,ð=ç �¹åÃì�âäñnâ�à�ìºð�æVñnñníVò{ß,à�â�ßÃî?ô¾íVèÏêyâäãÁñníVß,àÁâºß¹ò{â�àÁâºç[ã:âºß=àÁáÃç$êyâ��Tçnè�çnßÃñnç»ã�ñ9á{çnó�çV÷�Þ¤ß ���� � ��ç �Rçeþ>òÃçæ�ßÃê ��â�æ�å{åÃì�ðBàÁáÃçnâºè�ó�çnàÁá{íyêjàÁí&ã²í>ì�ë>çFü,à�íVýVçeã7ötíÊé é`â�à�á�ç[ìºæ>ã¤à�âäñ?ïtí>ò{ßÃê{æVèÁâºç[ã�íVè?ãÁò{è²ô�æ>ñ�ç/àÁçnßtã²âºíVß�÷�»íÊé�çnëVç[è � àÁí�íVòÃè�ý¹ß{íÊé`ìºç[ê{îVç/æVßpæVêÃæ�åyà�â�í>ßEí�ô�à�á{ç=ó�ç�à�á{íyêEô¾íVè�àÁáÃçFô¾òÃì�ì^û$æPë,âºç[è²õ¤ü,àÁí>ýVçeãÏç[þ,òÃæÊà�âºíVßáÃæVãÏß{íVà`ïtç[çnß&ò{ßÃêyç[è²à9æ�ý>çnß�ðVç�à[÷Þ¤ß=à�á{âäã^åÃæ�å]çnè é-ç`å{è�ç[ãÁçnß,à æ(ãÁç[ñníVßÃê�íVè9êyç[èRß,òÃó�çnè�âºñ[æ�ì,ó�ç�à�á{íyê/ô¾í>èRàÁáÃç7âºßÃñ�í>ó�å{èÁçeãÁãÁâºï{ìºçRû�æPë¹âºçnèÁõ

ü,àÁí>ýVç[ã-ç[þ,òÃæ�àÁâºíVßIô¾íVè�à¤é-íVõhåÃáÃæVãÁç7öÃíÊéqé`â�à�ágãÁò{è²ô�æ>ñ�ç�àÁç[ßÃã²âºí>ß÷! $â��]ç[èÁç[ß,àÁìºð�ô¾èÁí>ó å{èÁç[ë¹â�í>òÃã-ó�ç�àÁáÃí¹êÃãã²í>ì�ë¹âºß{î=à�á{ç�û�æPë¹âºçnèÁõ¦ü¹àÁíVý>ç[ã`çeþ,òÃæÊà�â�í>ßjíVß ��yç[êpîVè�âäê{ã � çV÷ îÃ÷ � �� ����� � ��� ��� � �������� �������!� é-çFàÁè�ç[æ�ààÁá{çBâ�ß,à�çnèÁô�æVñ�çBæVã/æpã²átæ�è�å6êyâäãÁñníVß,àÁâºß¹ò{â�à¤ð�÷&ú ç�æ>ñ9á{âºçnë>ç=àÁá{âäã�ï¹ð ç��yå{ìºâäñ�â�à�ì�ð�â�ßtñ�íVè�å]íVè9æÊàÁâºß{î � ò{ó�åÃãæVñ�è�í>ã�ã àÁá{ç�âºß,àÁç[è²ô�æ>ñ�ç$âºß>à�í/à�á{ç�êyâäã�ñ�è�ç�àÁç?êyâ��Tçnè�çnßÃñnç$í>åtç[è�æ�àÁí>è�ãn÷��Ïá{ç?íVè�î>æ�ßÃâ��eæÊà�â�í>ß�í�ô:àÁá{âäãÏåÃæVåtç[è`âäãæVã-ô¾íVìºìºíÊé7ã#")Þ¤ßgà�á{ç(ã²çeñ�àÁâºí>ß � é-ç$å{è�ç[ãÁçnß,àÏàÁá{ç?îVíÊë>çnè�ß{âºß{î(ç[þ,òÃæ�àÁâºíVßtã-æVßÃêIà�á{ç(æVãÁð¹ó�åyàÁíVàÁâäñ7æ�ßtæ�ìºðyã²âäãâ�ßIàÁá{ç?ìäæ�è�îVç»ê{çnßÃãÁâ�à¤ð�è9æÊà�âºí�ìºâ�ó�â�à[÷�Þ¤ßIà�á{ç$ô¾íVìºìºíÊé`â�ßÃî?ãÁç[ñ�àÁâºíVßgé-ç$îVâºëVç7àÁá{ç(êyçnà�æ�âºìäã�íVô)íVòÃè�ß¹ò{ó�ç[èÁâäñ[æ�ìó�ç�àÁáÃí¹ê�æ�ßtêFâºß=ã²çeñ�à�â�í>ß � é�ç7ã²áÃíÊé6è�ç[ãÁò{ì�à�ã^æ�ßÃê=ñníVß¹ëVç[èÁî>çnßÃñnç�ã²àÁòÃê{â�çeã^í�ôTß¹ò{ó�çnè�âäñnæ�ì,ç��yå]çnè�âºó�çnß,à�ãn÷

Page 4: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

� �+Ù����`Ø�ÖEÝ[Ö����gÛ��$×:ÝnÙFÖ��ú�ç?æ�è�ç�ìºí¹íVý¹âºß{î?ô¾í>èÏæFß¹ò{ó�çnè�âäñnæVì{ó�ç�à�á{íyê�àÁí=ãÁíVìºëVç»àÁá{ç$âºßÃñ�í>ó�å{èÁçeãÁãÁâºï{ìºçÏû�æPë¹âºçnèÁõ¦ü,à�íVý>ç[ãRçeþ>òtæÊàÁâºí>ßÃãô¾íVè-à¤é-íVõhå{átæVãÁç`à¤é-íVõ´ê{â�ó�ç[ßÃãÁâ�í>ßÃæ�ì,öÃíÊé é`â�à�áBãÁò{è²ô�æ>ñ�ç`à�çnßÃãÁâºíVß÷^Þ¤ßBç[æVñ9áIöÃò{âäê�å{áÃæ>ã²ç �� � í>è ��� ��àÁá{çó�íVó�çnß,à�ò{ó æ�ßÃê�óIæVã�ã²õhïÃæVìäæ�ßÃñnç7ç[þ,òÃæ�àÁâºíVßÃã�æ�è�ç

�������� � ����� � � � � �� ��� � � ��!"�$#&%('() � � �� ��� � �+* �!� �

�»çnè�ç �"� � , ).- �0/ âäã»à�á{ç�ëVç[ìºí¹ñnâ�à¤ð � � àÁá{çIê{çnßÃãÁâ�à¤ð � � àÁáÃç�å{èÁçeãÁãÁò{è�ç � æVßÃê !1�32+45�62 �87 ���:9 �7 ���&9 / �`à�á{ç=ã¤à�èÁçeãÁã»àÁç[ßÃã²í>è�é`â�à�á � �;�<4 � � � = � � ÷ �Ïá{ç�îVè9æPë¹â�à9æÊà�â�í>ßÃæ�ìTô¾í>è�ñnçFæ>ñ�à�ã�é`â�àÁá�ñ�íVßtã¤à9æ�ß,à# âºß�àÁá{ç�êyâºèÁçeñ�àÁâºí>ßBíVô:à�á{ç�ò{ßÃâ�à�ëVç[ñ�àÁí>è % ' ÷ �Ïá{ç�ï]íVò{ßtê{æ�è�ð�ñníVßÃê{â�à�âºíVßÃãÏæÊà�àÁá{ç/â�ß,à�çnèÁô�æVñ�ç(ï]ç�à¤é�çnç[ßà¤é-í�âºó�ó�âºãÁâºï{ìºçÏë,âäã�ñ�í>òÃã-öÃò{âäê{ã`æ�è�ç$àÁá{ç(ñníVß,àÁâºß¹ò{â�à¤ð�í�ô�àÁá{ç(ë>çnìºíyñ�â�à¤ð

� � � � � � � ) � � �

æ�ßÃêgàÁá{ç�ê{ð,ßtæ�ó�âäñ7ï]íVò{ßtê{æ�è�ðgñ�í>ßÃêyâ�àÁâºí>ßgïÃæVìºæVßÃñ�âºß{î�àÁá{ç(ß{í>èÁóIæVìæ�ßÃêgà�æVß{îVç[ß>à�âäæ�ìã¤à�èÁçeãÁãÁç[ã�"� � � � � �?>@� � � A 4 � � �B> � CEDE) � � �F � � � A 4 � � �B> � � ) � � �

é`á{çnè�ç C ê{çnß{íVàÁç[ãFà�á{ç�ãÁò{èÁô�æVñnçgà�çnßÃãÁâºíVß ñ�í¹ç���ñnâºçnß,à � D àÁáÃç�ñnò{èÁëÊæ�àÁò{è�ç � > à�á{ç�ß{íVè�óIæ�ì-ëVç[ñ�àÁí>è�æ�ààÁá{ç=âºß,àÁçnèÁô�æVñnç/å]íVâºß,àÁâºß{îgô¾èÁí>ó öÃò{âäê �� �$âºß,àÁíBöÃò{âäê ��� � � æ�ßtê F à�á{ç=à�æVß{îVç[ß,àÁâäæ�ì:ëVçeñ�àÁí>èn÷Fú�ç=áÃæPë>çâ�ß,à�èÁíyêyòÃñnç[ê�á{ç[èÁç�à�á{çIß{íVà�æ�àÁâºíVß � � � � � � � � �HGJILK � � � �MGJNOK$ô¾í>èFæ�ß¹ð � ò{ó�å æVñnèÁí,ãÁã$à�á{çgâºß,àÁç[è²ô�æ>ñ�çV÷ �ÏáÃçñ�íVß,à�â�ß¹ò{â�à¤ðBí�ô�à�á{ç=ëVçnìºíyñ�â�à¤ð �ÃçnìäêEà�íVî>ç�àÁáÃçnè$é`â�à�áEàÁáÃç�ó�í>ó�çnß,àÁòÃó8çeþ>òtæÊàÁâºí>ßÃã»âºó�å{ìºâºç[ã�P PRQ �QTSEU U � ���é`á{çnè�çVQQTS �XWW S � �Y�H� ÷ �Ïá{ç[èÁçnô¾íVè�ç � é-ç$îVçnà�æFïtí>ò{ßÃê{æVèÁð�ñ�í>ßÃêyâ�àÁâºí>ß=ô¾íVè-àÁá{ç$å{è�ç[ã�ã²ò{è�ç»î>è�æ>êyâºçnß,à�âºßàÁá{ç?ô¾í>èÁó Z Z �� ���+[ [\� � � 2 = � � � � � �

é`á{âäñ9ágé`âºìºì]ï]ç�æ�ß�ç[ã�ãÁçnß,àÁâäæ�ìTâºß{î>èÁçeêyâºçnß,à ô¾í>è`íVò{è`ß¹ò{ó�ç[èÁâäñnæVìtó�çnàÁá{íyêT÷�tíVè�àÁáÃç$è�çnåÃèÁçeã²ç[ß>à9æÊà�â�í>ß�í�ô�àÁá{ç?ó�íÊë¹âºß{î(âºß>à�çnèÁô�æVñnç»é�ç?æVå{å{ìºð�æ/ìºç[ëVçnìTãÁç�à7æ�å{åÃèÁí,æVñ9á÷ �Ïá{ç$â�ß,à�çnèÁõ

ô�æVñ�ç(âäã`âºó�å{ìºâäñ�â�à�ì�ð�îVâºë>çnß�ï,ðBàÁáÃç �nç[èÁí�ìºçnë>çnìíVô^] ��_ )`� � � é`á{ç[èÁça]pâäã7êyç �Ãß{ç[ê�æVãÏà�á{ç/ãÁâ�î>ß{ç[ê�ß{í>èÁóIæVìêyâäã¤à9æ�ßÃñnç7àÁí�à�á{ç(âºß>à�çnèÁô�æVñnç�÷ �Ïá{ç$àÁâºó�ç�ç[ëVí>ì�ò{àÁâºíVß�íVô)àÁáÃç(ì�ç[ëVç[ì]ãÁç�à`ô¾í>ì�ìºíÊé7ã�àÁá{ç(æVê{ëVç[ñ�àÁâºíVßgç[þ,òÃæÊà�âºíVß

� ]���b� �c��� ] � �+* � � �

�Ïá{ç�ëÊæVì�òÃç�íVô�àÁá{ç?ìºçnë>çnìTã²çnà7æ�ìºìºíÊé7ã�æ/òÃß{âäþ>òÃç$êyâäã²àÁâºßÃñ�àÁâºíVß�ï]ç�à¤é�çnç[ßgà�á{ç$à¤é-íFöÃò{âäêIåÃáÃæVãÁç[ãn÷ �ÃíVè`æVß,ðþ>òtæ�ß,àÁâ�à¤ðed é-ç�ñ[æ�ß&êyç��tß{ç

d ��_ )f� � �3g d�GJNOK ��_ )f� � � â�ô ] ��_ )f� �:h �d�GiILK ��_ )`� � � íVàÁá{ç[èÁé`âäãÁç � � �

ú�ç7òtã²òÃæVìºì�ð/íVó�â�à^àÁá{ç$ã²ò{å]çnè9ã�ñ�è�â�å{à �� ��íVè ��� �^íVôàÁá{ç7öÃò{âäê=å{áÃæ>ã²ç � á{íÊé�çnë>çnè � ô¾èÁí>ó �!� �^â�à âäã�æVìºé�æPðyãñ�ìºç[æVè à�í�é`á{âäñ9ágå{áÃæ>ã²ç(æVß,ðBþ>òtæ�ß,àÁâ�à¤ð�ï]çnìºíVß{î,ã�÷j&kRl m$n�oqpsrutsoEvxw�yzw�p3ou{&|�}�~.yzr�w��@yzw1~.n�oq�zyz�6y�~�rs��}�n�yzp�n���oEw���y�~<��vx}�~.yzrÞ¤ß�à�á{âäãIã²àÁòÃêyð é�çjæ�è�çEåÃæ�èÁàÁâäñ�òÃìºæVèÁìºð�âºß>à�çnè�ç[ã²àÁçeê6âºß à¤é-íVõhåÃáÃæVãÁç�öÃíÊé7ã�é`â�àÁáAá{âºîVá êyç[ßÃãÁâ�à¤ð6è�æ�àÁâºí>ãn÷�Ïá{ç(æVãÁð,ó�å{àÁí�à�âäñ»æ�ßtæ�ìºðyã²âäã�é�ç?å{è�ç[ãÁçnß,à�áÃçnè�ç$âäã�ó�í�à�âºëPæ�àÁçeêIï¹ðIå{è�íVï{ìºç[óIã�é�ç$âºß{â�à�âºæVìºì�ðFçnßÃñníVò{ß,à�çnè�ç[ê

Page 5: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

ã²í>ì�ë¹âºß{î�àÁáÃç à�á{çní>èÁçnàÁâäñnæVì�ìºð`ç �,à�èÁç[ó�çnìºð�ãÁâºó�å{ìºçRå{è�íVï{ìºçnó�í�ôÃæ�ã¤à9æÊà�âºñ èÁí>ò{ßÃê(ïÃò{ï{ï{ìºç é`â�à�á�á{âºîVá�êyç[ßÃãÁâ�à¤ðè�æ�àÁâºí>ã � ãÁçnç(ãÁç[ñ�àÁâºíVß � ÷ � ÷��)í�âºß¹ëVçeã¤à�âºî>æÊà�ç`àÁáÃç(ç[þ,òÃæÊà�âºíVßÃã�âºßgà�á{ç(á{âºîVá�êyçnßtã²â�à¤ð�è9æÊàÁâºí�ìºâ�ó�â�à

� � � GJNOK� GJI�K � � )é-ç�óIæ�ýVç»àÁá{ç$ç[þ,òÃæ�àÁâºíVßtã êyâºó�çnßtã²âºíVßÃì�çeãÁã^é`â�à�áBãÁíVó�ç7è�ç�ô¾ç[èÁç[ßÃñ�ç�ì�ç[ß{î�à�á������� � à�á{ç$èÁçnô¾çnè�çnßÃñnç�ë>çnìºíyñ�â�à¤ð, ���� � æ�ßÃê�àÁá{ç�êyç[ßÃãÁâ�à¤ð&íVô àÁáÃç�áÃç[æPë¹ð�öÃò{âäê � GJI�KÁ÷ �Ïá{ç�å{è�ç[ã�ã²òÃèÁç�áÃæVã$ï]çnç[ßpß{í>ßyõ¦êyâºó�çnßÃãÁâºíVßÃæVì�â��[ç[êé`â�à�á � GJI�K ,����� æ�ßtê�àÁáÃç?àÁâºó�ç»é`â�àÁá���� � ��� , ����¹÷��tíVè�àÁáÃç(ó�íVó�çnß,à�ò{ó!ç[þ,òÃæÊà�âºíVßÃã ô¾íVè�àÁáÃç(ì�âºî>á>à öÃò{âäê�� �ÏæVßÃêBàÁá{ç(áÃç[æPë¹ð=öÃò{âäê ��� �`é-ç?îVçnà� � GJNOK� � � � � � ��� GiNOK � ���� GJNOK = � GJNOK � ���� %(' ) � �� � GJI�K� � � � ��� GiILK � ���� GJI�K = � GJI�K � ���� %.'�) � ��� �

é`â�à�á àÁá{ç��7çnð¹ß{í>ìäê{ã¤õ´ß¹ò{ó�ïtç[è�ã���� GJN�� ILK � , � � ���������� 2 GJN�� I�K ��!#"�æVßÃê�àÁáÃç �tèÁí>òÃêyçIß¹ò{ó�ïtç[è$��� �, ���� � # � ���� � !#" % ÷�ú çIæVã�ã²òÃó�ç�àÁátæÊà(àÁáÃç&�7çnð¹ß{í>ìºêÃã¤õ´ß¹ò{óFï]çnè�í�ô�àÁá{ç=ìºâºîVá,à$öÃò{âäê�è�çnóIæVâºßÃã(' � � �?æVã� � � ÷RÞ¤ß,à�èÁíyêyòÃñnâºß{î/à�á{ç�æVãÁð¹ó�åyàÁíVàÁâäñ7ç �yåÃæ�ßÃãÁâºíVßÃã� GJNOK � � GJN*) +MK � � � GiN*) " K � *B*B*� GJI�K � � GJI,) +fK � � � GJI,) " K � *B*B*� GJNOK � � GJN*) +MK � � � GJN*) " K � *B*B*� GiILK � � GJI-) +MK � � � GJI,) " K � *B*B* � ��� �

â�ßBç[þ,òÃæ�àÁâºíVß � ��� ��ð¹â�ç[ìäê{ã�à�á{ç?ô¾íVìºìºíÊé`âºß{î/ãÁç�à`í�ô^å]çnèÁàÁò{è�ïÃæÊà�âºíVßgçeþ,òÃæÊà�â�í>ßÃã�"� !#" " � � ��� GJN*) +MK� + " � � GJN*) +MK� � � ����� GJN*) " K � ���� GiN*) +fK = � GJN*) +MK � ���� %('� � GJI,) +fK� � � ����� GJI,) +fK � ���� GJI,) +fK = � GiI-) +MK � ���� % '� " � � GJN*) " K� � � ����� GJN*) K � ���� GiN*) " K = � GJN*) " K� � GJI,) " K� � � ����� GJI-) " K � ���� GJI,) " K = � GJI,) " K *

� ��� �

�Ïá{ç$ê{ð,ßtæ�ó�âäñ`ï]íVò{ßÃêÃæ�è�ðIñ�íVßtêyâ�àÁâºíVßIæ�à�à�á{ç?âºß>à�çnèÁô�æVñnç7å{è�ç[ãÁçnß,à�ã�â�à�ãÁçnì�ô�âºßIà�á{ç$æ>ã²ð¹ó�åyà�í�à�âºñ`ô¾è�æVó�ç�æVã� + " P P � G.+fK U U � >@� 4 GJI,) +MK��� GJILK � > � �/0� D��1 " P P � G 1 K U U � >@�32 4 GJI-) 1 K��� GJI�K � 4 GJN*) 1 !#" K��� GJNOK54 � > �*6 �

� ��� �

é`á{çnè�ç�/0� � C � � � � � �������� ,����� � è�çnåÃèÁçeã²ç[ß>à9ã-àÁá{ç?ú çnï]çnèÁõhß¹ò{ó�ï]çnè[÷ �Ïá{ç�ñníVß,àÁâºß¹ò{â�à¤ð=í�ô�àÁá{ç(ë>çnìºíyñ�â�à¤ðâ�ßgàÁá{ç(ë¹âäã�ñ�íVòtã�ñ[æVãÁç?å{èÁíÊë¹âäêyçeã òÃãÏé`â�à�á&æ�ï]íVò{ßtê{æ�è�ðgñ�í>ßÃêyâ�àÁâºí>ß�ô¾íVèÏà�á{ç(å{è�ç[ã�ã²ò{è�ç$îVè9æVêyâºçnß,à "� !#" " ��� GJN*) +MK � �� + " ��� GJN*) " K � ��� GJI,) +fK � 7 7 ���� = � G.+MK98 8� " " ��� GJN*) K � ��� GJI,) " K � 7 7 ���� = � G " K 8 8 * � � � �

�Ãò{èÁàÁá{ç[èÁó�í>èÁç � é�ç�áÃæPëVçgæ�à�çeæVñ9á áÃâ�ç[è�æVè�ñ9áÃâºñ[æ�ì^ìºçnë>çnì íVô7àÁáÃçBå]çnèÁàÁò{è�ïÃæ�àÁâºíVß~çeþ>òtæÊàÁâºí>ßÃã/êyâºë>çnè�îVçnßtñ�çô¾èÁç[ç`ëVçnìºíyñ�â�à¤ð"�Ãç[ìäê{ã�÷ �Ïá{ç7óIæVâ�ß=ñ9áÃæVè�æ>ñ�àÁç[è^í�ôTàÁá{ç»åtç[è²à�ò{èÁïtæÊàÁâºí>ßFçeþ>òtæÊàÁâºí>ßÃã^âäã:àÁáÃæ�à â�à�âºã�åtí,ãÁãÁâºï{ìºç à�í

Page 6: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

ã²í>ì�ë>ç�ã¤à�çnåjï¹ðEã²àÁçnå�ô¾íVè$ç[æ>ñ9áEå{áÃæVãÁçFãÁçnåtæ�è9æÊàÁç[ìºð�÷ �Ïá{ç=ñ�í>ßÃêyâ�àÁâºí>ß�íVô êyâºëVç[èÁî>çnßÃñnç?ô¾èÁç[ç/ëVç[ìºí¹ñnâ�àÁâºç[ã`æ�àç[æVñ9áEìºç[ëVçnì:è�ç[ãÁò{ì�à�ã7âºßEùRí>âºã�ãÁíVß&ç[þ,òÃæ�àÁâºíVßÃã7ô¾íVè�àÁá{ç�è�ç[ãÁåtçeñ�àÁâºë>ç�å{è�ç[ã�ã²ò{è�ç[ãn÷#»å{å{è�íVå{è�âäæÊà�ç�ïtí>ò{ßÃê{æVèÁðñ�íVßtêyâ�àÁâºíVßÃã�æ�è�ç�àÁí�ï]ç�êyçnè�âºëVç[êBã²òtñnñ�çeãÁãÁâºëVç[ì�ðFô¾è�íVó à�á{ç(á{âºçnè9æ�è9ñ9á¹ðFíVô:âºß,àÁç[è²ô�æ>ñ�ç?ñ�íVßtêyâ�àÁâºíVßÃã � ��� �»÷

��í�ì�çeæVêyâºß{î�í>è�ê{çnè � à�á{ç=å{èÁçeãÁãÁò{è�ç � GiN*) +fK`âºßEà�á{ç�ì�âºî>á>à�å{áÃæVãÁç �� �»âäã�ñníVßÃã²à�æVß,à$æ�ßtêEàÁáÃç/öÃòÃâºê�âäãåÃæVã�ã²âºë>çnìºð�ô¾íVìºìºíÊé`â�ßÃî�àÁáÃç/ó�íÊëVç[ó�çnß,àÏí�ô^àÁá{ç/á{ç[æPë¹ðIötò{âäêT÷��(ß{íÊé`âºß{î�àÁá{ç/ñníVßÃã²à�æVß,à7å{èÁçeãÁãÁò{è�ç � GJN*) +MKé-ç ñ[æ�ß/ãÁíVìºë>ç^ô¾íVè � GJI,) +fKæVßÃê � GiI-) +MKTé`â�àÁá $â�è�âäñ9á{ìºç�àï]íVò{ßÃêÃæ�è�ð?ñníVßÃê{â�à�âºíVßÃã]ô¾í>è)à�á{ç å{è�ç[ã�ãÁò{èÁç ùRíVâäãÁãÁíVßç[þ,òÃæÊà�âºíVßEîVâºë>çnß&ï¹ð � ��� �9÷ �Ïá{ç�ß{ç��¹à(ã¤à�çnå�âºßEà�á{ç�á{âºçnè9æ�è9ñ9á{âäñnæ�ìãÁðyã¤à�çnó âäã`à�á{ç�ãÁíVìºòyà�âºíVß�ô¾íVè � GJN*) +MKæ�ßÃê � GJN*) " K�é`â�àÁá ñ�í>ßÃêyâ�àÁâºíVßtã�íVß�àÁáÃç�åÃèÁçeãÁãÁò{è�çBî>è�æ>êyâºçnß,à ��� GJN*) " K/æVìºíVß{îEà�á{çEâºß,àÁçnèÁô�æVñnçgå{è�íÊë¹âäêyç[êï,ð&çeþ>òtæÊàÁâºí>ß � ��� �9÷(û�í�àÁçFàÁátæÊà$àÁá{ç�ãÁíVìºòyàÁâºí>ßBô¾í>è$àÁá{çFöÃíÊé í�ô-å{áÃæVãÁç áÃæ>ã�æBîVè9æVê{â�ç[ß,à»ïtí>ò{ßÃê{æVèÁðñ�íVßtêyâ�àÁâºíVßBíVß{ìºð�÷ �Ïá{âäãÏâºßÃêyâäñnæ�àÁçeã�àÁáÃæ�à�æ�å{è�íVå]çnè7âºó�å{ìºç[ó�çnß,à�æ�àÁâºíVß�íVô�à�á{ç/å{è�ç[ã�ãÁò{èÁç?î>è�æ>êyâºçnß,à � ò{ó�åñ�íVßtêyâ�àÁâºíVßÃã âäã-æVïÃã²í>ìºòyàÁç[ì�ð�ß{çeñ�ç[ã�ã�æ�è�ð/à�í�í>ïyà�æVâºßgæ�ß¹ð�ã²í>ìºòyàÁâºíVß�âºßgàÁá{ç?ìºíÊé�êyçnßtã²â�à¤ð=è�æ�àÁâºí�ìºâºó�â�àn÷ �ÏáÃçã¤à�è�æ�àÁçnî>ð�ñníVß,àÁâºß¹ò{çeã»é`â�àÁá � GJN*) +MK � � GJN*) " K � � GiI-) " K � � GJI,) " K � � GJN*) " K � � GJN*) K � � �B� ÷Fû»í�à�ç/à�áÃæÊàé-ç?áÃæPë>ç?æVìºé�æPðyã� $âºèÁâäñ9á{ìºçnà-ï]íVò{ßÃêÃæ�è�ðgñ�í>ßÃêyâ�àÁâºíVßtã-í>ß � GJN*) 1 K æ�ßÃêBû»ç[ò{óIæ�ßÃßBï]íVò{ßtê{æ�è�ðgñ�í>ßÃêyâ�àÁâºí>ßÃãíVß � GJI,) 1 K�÷

�ÏáÃç�âºó�å]íVèÁà�æVß>à�â�ß{ô¾íVè�óIæÊàÁâºí>ß�à�í�ßÃí�àÁâäñnç á{ç[èÁç�âäã�à�áÃæÊà�æ�ìºì¹æPëÊæ�âºìäæ�ï{ìºç^âºßyô¾íVè�óIæÊà�â�í>ß(æÊà^àÁá{ç�âºß,àÁçnèÁô�æVñnçæ�è�ç�ßÃç[ñ�çeãÁã�æ�è�ð(à�í�ã²í>ì�ë>ç à�á{ç»æ>ã²ð¹ó�åyà�í�àÁâäñ-ç[þ,òÃæ�àÁâºíVßÃãn÷ �jí>ã²à�ß,òÃó�çnè�âºñ[æ�ìyã�ñ9á{çnó�çeã:ô¾íVèRà¤é�í�õ´å{áÃæ>ã²ç-öÃíÊéêyíIß{íVà$æ�åÃå{ìºð�æVß,ð&ñ�í>ßÃêyâ�àÁâºí>ß�í>ß&àÁá{çFå{è�ç[ã�ã²ò{è�ç�î>è�æ>êyâºçnß,à`àÁí�ã²í>ì�ë>ç$ô¾íVè�àÁá{çFå{èÁçeãÁãÁò{è�ç�÷ �Ïá{âäã»ó�ç[æ�ßtãàÁáÃæ�à�àÁá{çEý¹âºß{çnóIæ�àÁâäñIï]íVòÃßÃê{æ�è�ð6ñníVßÃêyâ�à�â�í>ß6í>ß�à�á{çEëVç[ìºí¹ñnâ�àÁâºç[ãFæÊà�àÁáÃçEâ�ß,à�çnèÁô�æVñ�ç&âäã�ß{í�àIç[ßyô¾íVè9ñ�çeêæ�ßÃêEãÁå{ò{è�âºíVòÃã`òÃß{å{á¹ðyã²âäñnæVì�åÃæVè�æ>ã²â�àÁâäñ$ñ�ò{è�è�çnß,à�ã�æ�è�ç(ë,âºèÁàÁòÃæVìºì�ð�òÃßÃæPëÊæ�âºìäæ�ï{ìºç�÷-ú�ç�ñ�í>ßÃñ�ìºòÃêyç?à�áÃæÊà�àÁá{çñ�íVòÃå{ìºâ�ßÃî(í�ôàÁáÃç�å{è�ç[ã�ãÁò{èÁç»îVè9æVêyâºçnß,à9ãRíVßIï]í�à�áBãÁâäêyç[ã í�ô�à�á{ç�âºß,àÁç[è²ô�æ>ñ�ç7ãÁá{íVòÃìºêIïtç�â�ßtñ�íVè�å]íVè9æÊàÁçeê�â�ßBæß,òÃó�çnè�âºñ[æ�ì�ó�çnàÁá{íyê&ô¾í>è$à¤é-íVõhå{átæVãÁç(öÃíÊé é`â�àÁápá{âºîVá�êyç[ßÃãÁâ�à¤ð�è9æÊàÁâºíÃ÷ �tò{è²à�á{çnè�ó�íVè�ç � àÁáÃç�ß¹ò{ó�ç[èÁâäñ[æ�ìó�ç�àÁáÃí¹ê�ãÁá{íVò{ìäêFèÁçeêyòÃñnçÏàÁí�ãÁíVìºë¹â�ßÃî7àÁáÃç`ì�çeæVêyâºß{î$íVè9êyç[è^æ>ã²ð¹ó�åyà�í�à�âºñ-ç[þ,òÃæ�àÁâºíVßtã�â�ô]àÁá{ç7ê{çnßÃãÁâ�à¤ð�í�ô]àÁá{çì�âºî>á>àÏöÃòÃâºê&âäã7ãÁç�à»àÁí��[çnè�í{÷ÏÞ¤ß�ñ�í>ß,àÁè9æVã²à`àÁí�à�á{çFêyç[ëVç[ì�í>å{ó�çnß,à`í�ô^à�á{ç/âºó�ó�çnè9ãÁç[êIâºß,àÁç[è²ô�æ>ñ�ç?ó�ç�à�á{íyê� ��� ���� �� � íVò{èÏæ�å{åÃèÁí,æVñ9á�àÁí�ïtç$å{è�ç[ãÁçnß,àÁçeê�âºß�àÁáÃç»ô¾í>ì�ìºíÊé`âºß{î?ã²çeñ�à�â�í>ßÃã�âäã-ìºç[ã�ã�ó�í�à�âºëPæ�àÁçeê=ï,ð�à�á{ç�î>í>æVìàÁí�âºßÃñnèÁçeæVãÁçÏàÁá{ç$æ�å{å{è�í��yâºóIæÊàÁâºí>ß(íVè9êyçnè-í�ôTàÁá{ç�ó�ç�à�á{íyê=ï{òyàÏè�æ�àÁá{ç[è�ï¹ðFà�á{ç»ßÃç[ñ�çeãÁãÁâ�à¤ð�à�í�âºßÃñ�ìºòÃêyç7æVìºìèÁç[ì�ç[ëÊæ�ß,à-å{á¹ðyãÁâºñ[æ�ìTâºßyô¾íVè�óIæÊà�âºíVß�í�ô:àÁáÃç(ò{ßÃêyç[èÁìºð¹âºß{î/å{è�íVïÃì�ç[ó&÷

� �HÛ�� �`Ø�Ý[Ü �������Ï×EÙgÚ�ÃíVè»àÁá{ç�ß¹ò{ó�çnè�âäñnæVì�ã²í>ì�ò{àÁâºíVß&í�ôRà�á{ç/ó�í>ó�çnß,àÁòÃó8çeþ>òtæÊàÁâºí>ß � � ��ã²òÃï � ç[ñ�à»àÁígàÁá{ç�âºßÃñ�í>ó�å{è�ç[ã�ã²âºï{âºìºâ¶à¤ðñ�íVßtã¤à�è�æVâ�ß,à �!� �Ïé�ç�òÃãÁç/æ�å{è�í � çeñ�àÁâºí>ßBó�çnàÁá{íyê&íVßjñ[æ�èÁàÁçeã²âäæ�ß�îVè�âäê{ãn÷�ù-èÁí � ç[ñ�à�âºíVß�ó�ç�àÁáÃí¹êÃãÏô¾íVè»à¤é-íVõå{áÃæVãÁçIâºßÃñníVó�å{è�ç[ã�ã²âºï{ìºç�æ�ßÃê~ëÊæVèÁâäæ�ïÃì�ç�êyç[ßÃãÁâ�à¤ð�ìºíÊé ��æVñ9áyõ´ß¹ò{óFï]çnè�öÃíÊé áÃæPëVçgïtç[çnß å{è�ç[ãÁçnß,à�ç[ê~âºßã²ç[ëVçnè9æ�ìyåtæ�å]çnè9ã�å{è�çnë¹âºíVòÃãÁìºð � çV÷ îÃ÷ � � � � ����� ���� ������� ��� ��� ÷ �ÏáÃçnð�æ�è�ç7æ�ìºìtãÁç[ñníVßÃêyõhí>è�êyç[èRó�ç�à�á{íyê{ã æVßÃêñnæ�ß=ï]ç7à�è�æ>ñ�ç[ê�ïÃæ>ñ9ý/à�í/ã�ñ9á{çnó�çeã^í�ô ��æVèÁìºíÊé�æVßÃê�ú çnìäñ9á � ��� � í>èRàÁáÃç7å{è�í � ç[ñ�àÁâºíVß=ó�çnàÁá{íyê�í�ô�ÿ-á{í>èÁâºß� ����� ÷ �Ïá{ç[ãÁç(ó�ç�à�á{íyê{ã`ó�çnàÁá{íyê{ã`átæPëVç?âºß&ñníVó�ó�í>ßIàÁátæÊà7à�á{çnð �Ãè�ã²à7òÃãÁç/æ�ß&ç �¹åÃì�âäñnâ�à`êyâäã�ñ�è�ç�àÁâ��eæÊàÁâºí>ßô¾íVèFàÁá{ç�ç��Tç[ñ�à�ã�í�ô�ñníVß¹ëVçeñ�à�â�í>ß â�ß�æ�å{è�ç[ê{âºñ�àÁí>è/ã¤à�çnå � æ�ßÃê~àÁá{ç[ß6çnß{ô¾íVè9ñ�çBàÁá{ç&ñ�í>ó�å{ìºâºæVßÃñ�ç=é`â�àÁá�æëVçnìºíyñ�â�à¤ð&êyâºëVç[èÁî>çnßÃñnçFñ�í>ßÃã²àÁè9æ�âºß,à$âºß æ�ãÁò{ïÃãÁç[þ,ò{ç[ß,à?åÃèÁí � ç[ñ�à�âºíVßpã²àÁç[å÷ �Ïá{ç[ð�êyâ��]ç[è(âºß�à�á{çnâºè?òÃãÁç�í�ôñ�íVßtã²ç[èÁëÊæÊà�âºëVç � � ����� ß{í>ßyõ´ñníVßÃãÁçnè�ëÊæÊà�â�ë>ç � � � �����$���� ����� ãÁñ9á{ç[ó�ç[ãÏô¾í>è7à�á{ç/ó�í>ó�çnß,àÁò{óÑç[þ,òÃæ�àÁâºíVßEæVßÃêâ�ßgàÁá{ç(òtã²ç(íVô�ç��{æ>ñ�à ���� � ��� � � ��� � ��� � íVè7æ�åÃå{èÁí��yâºóIæÊà�ç � ������ å{èÁí � ç[ñ�àÁâºíVßÃãn÷

ø�òÃè/ß¹ò{ó�çnè�âäñnæVì:ó�çnàÁá{íyê âäã�ãÁâºó�â�ìäæVè7àÁíjàÁá{çeã²ç�ãÁñ9á{ç[ó�ç[ã?ç �yñnçnåyà/ô¾íVè�à�á{çgâºßÃñníVè�åtí>è�æ�àÁâºíVß�í�ôÏàÁá{çïtí>ò{ßÃê{æVèÁð&ñ�í>ßÃêyâ�àÁâºíVßtã7æÊà�àÁá{ç�âºß>à�çnèÁô�æVñnç�ïtçnà¤é-ç[çnßEà�á{çFà¤é-í�ötò{âäê{ã�÷ �Ãí>ì�ìºíÊé`âºß{î � � � � � ����� é-ç�æ�å{åÃì�ðà¤é-ípå{èÁí � ç[ñ�àÁâºíVßÃã�åtç[èFà�âºó�ç[ã²àÁçnå "IíVß{çgô¾íVè�àÁá{ç�âºß,àÁçnèÁô�æVñnçIëVç[ìºí¹ñnâ�àÁâºç[ã(æVßÃê~à�á{ç�ãÁç[ñníVßÃê6íVß{çgô¾íVèFàÁá{çñ�çnìºì�õ¦ñ�çnß,à�çnè�ç[ê�ëVçnìºíyñ�â�à�â�çeã�÷ �ÃíVèBàÁáÃç�ãÁåÃæÊà�âäæ�ì�ê{âºã�ñ�è�ç�à�â��[æÊà�âºíVß6é�ç�òÃã²ç�àÁá{ç ã²à�æ�î>îVç[èÁçeêAîVè�âºê ãÁðyã¤à�çnóâ�ß,à�èÁíyêyòÃñnç[êEï¹ð �$æ�è�ìºíÊé æ�ßÃê�ú�ç[ìäñ9á ��� � � ÷ �^âºîVòÃèÁç � ãÁá{íÊé7ã?æ�îVè�âäêjñ�ç[ì�ì^é`â�àÁápàÁáÃç�âºßÃê{ç��jß{íVà�æÊà�âºíVßæ�ßÃêgàÁá{ç(ìºíyñnæ�àÁâºíVßgí�ô:àÁá{ç?ëÊæ�è�âäæ�ï{ìºç[ã-òÃãÁç[êBâºßgà�á{ç(ß¹ò{ó�çnè�âäñnæ�ì]ó�ç�à�á{íyêT÷

Page 7: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

���������������� �� � � ���

� � � ����� � �

������ �� ���

��������� ����������� � ���

������ � � ��� ����������� ��� ���������� �� � ��� � � �����������

�:âºî>ò{èÁç � "! $èÁâäêBñ�çnìºì]é`â�àÁágâºßÃê{ç��Iß{íVà�æÊà�âºíVßIæ�ßtêgìºíyñnæ�àÁâºíVßÃã�í�ô)à�á{ç$ëPæVèÁâäæVï{ìºç[ãn÷�"#�âäã-àÁá{ç(ñnçnìºì]ñnçnß,àÁç[è �æ�è�ç»à�á{ç�ñ�ç[ì�ìâºß,àÁç[è²ô�æ>ñ�çeã � æ�ßtê%$Fæ�è�ç$àÁá{ç(î>èÁâäêgß{íyêyç[ãn÷&&kRl ' r��6oEw&~(|�� ou{&|�}L~.yzrswÞ¤ß>à�çnî>è�æ�àÁâºß{î�ç[þ,òÃæÊà�âºíVß � � �Ïâºß&àÁâºó�ç(ïtçnà¤é-ç[çnß �&� �)( æVßÃê �&� �)(+* " é-ç/áÃæPëVç?ô¾í>è7àÁáÃçFñ�ç[ìºì�õ¦ñ�ç[ß,àÁçnè�ç[êëVçnìºíyñ�â�à�â�çeã �!, ) - àÁí�ã²çeñ�í>ßÃêgí>è�ê{çnè`âºßIà�â�ó�ç�"

� (.* ", ) - � � (, ) - � = � /0 ��� �c� � � � � (+*213, ) - � �� (+* 13, ) - � ��� � (+*213, ) - �c2 (+*213, ) - � = � � (+*213, ) - � #+% '�45

� �!6, ) - � = �� (+*713, ) - � ��� � (+*713, ) - * � �� �

�Ïá{çBß¹ò{ó�ç[èÁâäñ[æ�ì æVì�î>íVè�â�àÁá{ó à�í�æ>êyëÊæ�ßÃñnçIà�á{ç&ëVçnìºíyñ�â�à�â�çeã$ô¾èÁí>ó àÁâºó�ç �\� � ( àÁí �e� � (+* " ñnæVß ïtçã²ò{ó�óIæVèÁâ��[ç[ê�æVã-ô¾í>ì�ìºíÊé7ã�"� ÷�ÿ-íVó�åÃòyà�æ�àÁâºíVßjíVô-à�á{çIæ�ò�¹âºìºâäæ�è�ðgâºß,àÁç[è²ô�æ>ñ�ç/ë>çnìºíyñ�â�àÁâºçeã � 6 ,98 " % ) - 8 "% ï¹ð �:æPð¹ìºíVèÁõhç �¹åtæ�ßÃãÁâºíVß�í�ôàÁáÃç�ñ�çnìºì�ñnçnß,àÁç[èÁçeêgë>çnìºíyñ�â�àÁâºç[ã âºß�ãÁåÃæVñnç$æVßÃêBàÁâºó�ç�÷

� ÷`ù è�í � çeñ�à�â�í>ß»íVô>à�á{ç æ�ò�¹âºìºâäæ�è�ð-âºß,àÁç[è²ô�æ>ñ�ç^ëVç[ìºí¹ñnâ�àÁâºç[ã{àÁí7î>ç�à)êyâºëVç[èÁî>çnßÃñnç)ô¾è�çnçRë>çnìºíyñ�â�àÁâºçeã � (.*213,98 " % ) - 8 "% æÊà`à�á{ç�ñ�ç[ìºì]âºß,àÁç[è²ô�æ>ñ�çeã�÷� ÷�ÿ-íVó�åÃòyà�æ�àÁâºíVßBí�ô)à�á{ç�æ�ò�¹âºìºâäæ�è�ðFñnçnìºì�õ´ñnçnß,àÁç[èÁçeê�ëVç[ì�íyñnâ�à�âºç[ã � 6 ) (.* ", ) - òÃãÁâ�ßÃî � (+* 13,:8 "% ) - 8 " % � ÷`ù è�í � çeñ�à�â�í>ßEí�ôRà�á{ç�ñ�çnìºì�õ¦ñ�çnß,à�çnè�ç[ê�ëVç[ì�íyñnâ�à�âºç[ã�àÁí�îVçnà$êyâºë>çnè�îVçnßtñ�ç(ô¾è�çnç=ñ�ç[ì�ì�õ¦ñ�ç[ß>à�çnè�ç[êBëVç[ì�íyñnâ�à�âºç[ã� (.* ", ) - æÊàÏàÁâºó�ç$ìºçnëVç[ì �^� � (+* "

7éÏæPð~ô¾èÁí>óHàÁáÃçjâºß>à�çnèÁô�æVñnçBí>ò{è�ó�ç�à�á{íyê ñníVè�èÁçeã²å]íVßtê{ãFàÁí æ~ã²à�æ�ßtê{æ�è9ê�ãÁç[ñníVßÃê�íVè9êyç[è�å{èÁí � ç[ñ�àÁâºíVßó�ç�àÁáÃí¹ê÷ �»íÊé�çnë>çnè � æÊà»àÁá{ç�âºß>à�çnèÁô�æVñnç?àÁá{çFó�ç�àÁáÃí¹ê�êyâ��]ç[è�ã7ãÁâºîVß{â��tñ[æ�ß,àÁìºð�âºßjá{íÊé�é�ç/ñ�í>ó�å{òyà�ç(àÁá{çå{èÁçeãÁãÁò{è�ç&îVè9æVêyâºçnß,à9ã�æVßÃê àÁáÃçnè�ç�ô¾íVè�çEá{íÊéÑé-çpêyâäãÁñnèÁçnàÁâ��nçBàÁá{çpêyâäãÁñnèÁçnàÁç ��æVå{ìäæVñ�çnõhí>åtç[è�æ�àÁí>è/ô¾í>è�àÁá{çå{èÁçeãÁãÁò{è�ç�ùRí>âäãÁãÁíVßyõ´ç[þ,òÃæ�àÁâºíVß�÷;�<�=�<�= >@?BADCFEHGI�J.CLKMC�NPO�J+QRADQPCTS�Ïá{ç/âºß,àÁç[è²ô�æ>ñ�ç(ëVç[ìºí¹ñnâ�àÁâºç[ã`æVèÁç/ñníVó�å{ò{àÁç[ê&ï¹ðEæ �:æPð¹ì�í>è²õ´ç��yåÃæVßÃãÁâ�í>ßIâºß�ãÁåÃæVñnç/æ�ßÃêEàÁâºó�ç�í�ôRà�á{çFñnçnìºì�õñ�çnß,à�çnè�ç[ê�ëVç[ì�íyñnâ�à�âºç[ã?ñ�í>óFï{âºß{çeê�é`â�à�á æ�ß~ò{å¹é`âºßÃêyâºß{î&å{è�íyñ�ç[ã�ãn÷ �ãFæVß~ç��{æ�ó�å{ìºç � é-çgáÃæPëVçIæ�à�àÁá{çâ�ß,à�çnèÁô�æVñ�ç7U � " ) V � ã²ç[ç �:âºî>ò{èÁç ��� ô¾íVèÏàÁáÃç(ëVçnìºíyñ�â�à¤ð=ñ�í>ó�åtí>ß{çnß,à , ß{íVè�óIæ�ì]à�íFà�á{ç(âºß,àÁçnèÁô�æVñnç", (+* 13, * 13 ) - � , 6 , *213 ) - � �� , * 13 ) - 2 ����BW 4 (, *713 ) - * � ��� �

Page 8: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

�Ïá{ç�æ�ò��yâºìºâäæ�è�ðFë>çnìºíyñ�â�à¤ð , 6 , *213 ) - âºã�à�á{ç(à�â�ó�çnõ´æ>êyëÊæ�ßÃñnç[êgëVçnìºíyñ�â�à¤ð�é`â�àÁáÃíVòyà`à�á{ç/ñ�í>ß,àÁè�â�ïÃòyàÁâºíVßBí�ô:àÁá{çå{èÁçeãÁãÁò{è�ç�î>è�æ>êyâºçnß,àn÷:Þ¦à`âäã`êyç�à�çnè�ó�â�ßÃç[êIï¹ðgæ=ã²à�æVßÃê{æ�è9êBò{å¹é`â�ßtêyâºß{î/å{è�íyñ�çeêyò{è�ç"

, 6 , * 13 ) - ������� �����, 6 ) �, * 13 ) - ) â�ô , 6 ) �, * 13 ) -�� � æ�ßÃê�� , 6 ) �, * 13 ) - � , 6 ) , * 13 ) -��� �

, 6 ) , *213 ) - ) â�ô , 6 ) , *213 ) - h � æ�ßÃê � , 6 ) �, *213 ) - � , 6 ) , *713 ) -� h ��� � , 6 ) �, *213 ) - � , 6 ) , *213 ) -� ) í�àÁáÃçnè�é`âºãÁç *

�Ïá{ç�ëÊæ�ìºò{çeã]ô¾íVè�à�á{ç æVò��yâ�ìºâäæ�è�ðÏëVç[ì�íyñnâ�à�âºç[ãtíVß?àÁáÃç ìºç�ôäà)æVßÃê?èÁâºî>á>à)ãÁâäêyçRí�ô¹à�á{ç�âºß,àÁçnèÁô�æVñnçRæ�è�ç êyç�à�çnè�ó�âºß{ç[êï,ð ��æPð¹ìºíVèÁõhç �yåÃæ�ßÃãÁâºíVßÃã^âºß&ã²åtæVñ�ç(æVßÃêIà�âºó�ç�íVô�à�á{ç�ñ�ç[ì�ì�õ¦ñ�ç[ß>à�çnè�ç[ê=ë>çnìºíyñ�â�àÁâºç[ã

, 6 ) �, * 13 ) -� , (, ) - � � � ,�BW (, ) - = W� � � � ,��� 6 , ) - = �� ), 6 ) , * 13 ) - , (, * " ) - � � � ,�BW (, * " ) - = W� � � � ,��� 6 , * " ) - = �� ) � ��� �

é`á{çnè�ç?àÁá{ç�àÁâºó�ç�êyçnè�âºëÊæÊàÁâºë>ç[ã�êyíIß{íVà»âºßÃñ�ìºòÃêyç$à�á{ç�å{è�ç[ã�ã²òÃèÁç(î>è�æ>êyâºçnß,àn÷,�»çnå{ìäæVñnâºß{î � W��W S � 6 é`â�àÁá�àÁá{çó�íVó�çnß,à�ò{ó!ç[þ,òÃæÊà�âºíVß � � ��ð¹âºçnìäê{ã, 6 ) �, * 13 ) -� , (, ) - � � � ,�BW (, ) - = W

� � � , � � , �c2 =a, �Y# ��� � (, ) - = ��

, 6 ) , * 13 ) -� , (, * " ) - � � � ,�BW (, * " ) - = W� � � , � � , �c2 =a, �Y# � ��� (, * " ) - = �� * � ��� �

�ÃíVè$à�á{ç�ëVç[ì�íyñnâ�à¤ð&î>è�æ>êyâºçnß,à�ã$æ�ßÃê�àÁáÃç���æ�åÃìºæ>ñ�çFíVå]çnè9æÊàÁí>è?â�ß � ��� ��é�ç=æVå{å{ìºðjñ�í>ß¹ëVçnß,à�â�í>ßÃæ�ì:ã²çeñ�í>ßÃêíVè9êyçnè�ô¾íVè�ó�ò{ìäæVã � ç�÷ î{÷

2 � ,�BW 4 , ) - , , * " ) - � , , ! " ) -� = W æVßÃê � =a, � , ) - , , * " ) - � � , , ) - � , , ! " ) -= W *�ÃíVèFàÁáÃç�ñníVó�å{òyà9æÊà�â�í>ßÃã�âºß~à�á{âäãFã²àÁòtêyð~â�à�éÏæVãFß{í�à=ß{çeñ�ç[ã�ã�æ�è�ð�à�ípòÃãÁç�ì�âºó�â�àÁçeê�î>è�æ>êyâºçnß,à�ã�ô¾íVèFàÁá{çñ�íVß¹ë>ç[ñ�à�âºëVç7à�çnè�óIã�÷Þ¤ß�íVè9êyç[è-à�íFî>ç�à7æ�ß�ç[þ,òÃæ�àÁâºíVßIô¾íVè�àÁá{ç(åÃèÁçeãÁãÁò{è�ç»é�ç?æVå{å{ìºð�àÁá{ç?âºßÃñníVó�å{è�ç[ã�ã²âºï{âºìºâ�à¤ð(ñ�íVßtêyâ�àÁâºíVßIí>ß

àÁá{ç�â�ß,à�çnèÁô�æVñ�ç`ë>çnìºíyñ�â�àÁâºçeã)âºß�âºß,àÁçnî>è�æVìyô¾íVè�ó&÷��tíVè àÁá{ç$ñ�íVß,à�èÁí>ì{ëVí>ì�òÃó�ç�� , ) - é`â�àÁá�ìºçnßÃî�àÁá = W æ�ßÃê =��æ�è�íVò{ßÃêIàÁá{ç�ñnçnìºì�õ´ñnçnß,àÁç[è!U )V é-ç?áÃæPë>ç?â�ß�êyâäãÁñnèÁçnàÁç�ô¾íVè�ó�= W =����W������ � � (.* 13 �B> � � , (+* 13, * 13 ) - � , (+* 13, ! 13 ) -= W � - (+*713, ) - * 13 �$- (+*213, ) - !#" % =��

� � � ��� � (.* 13, ) - � �+* � �� ��à/àÁáÃâºã/åtí>âºß>à�â�à/âäã�âºó�åtí>è²à9æ�ß,à(àÁípß{í�à�âäñ�ç�à�áÃæÊà�â�ß6ó�â �yçeê�ñnçnìºìäã?àÁáÃçBêyâºëVç[èÁî>çnßÃñnçIñ�í>ßÃã¤à�è�æVâºß>à � �� �óFòÃã²à?â�ßtñ�íVè�å]íVè9æÊàÁçFëVç[ì�íyñnâ�à�âºç[ã`ô¾è�íVó+ïtíVàÁá�öÃò{âäêpå{áÃæ>ã²çeã �� �?æVßÃê ��� �9÷Iú�á{âäñ9á�âºß>à�çnèÁô�æVñnçFë>çnìºíyñ�â�à¤ðïtç[ì�í>ß{î>ã(àÁí�é`á{âäñ9á öÃò{âäê~å{áÃæ>ã²çgâºãFò{ß{âäþ,ò{ç[ì�ð�êyç��tß{ç[ê6ï,ð�à�á{çBëÊæVì�òÃçIí�ô`à�á{çBìºç[ëVçnì�ã²çnà�æ�à/àÁáÃç�ñníVèÁõèÁçeã²å]íVßÃê{â�ßÃî�âºß,àÁç[è²ô�æ>ñ�ç�ìºíyñnæ�àÁâºíVß÷! �ãÁâºß{îEç[þ,òÃæ�àÁâºíVß � ��� � � â�à/ô¾íVìºìºíÊé7ã»ô¾è�íVó � �� �?àÁá{ç�ã²ç[ó�â�õ¦êyâäã�ñ�è�ç�àÁçùRíVâäãÁãÁíVß�çeþ,òÃæÊà�â�í>ßIô¾í>èÏàÁá{ç(å{è�ç[ã�ãÁò{èÁç$å "

�= W#" �� , * 13 ) - 2 ����BW 4 , * 13 ) - � �� , ! 13 ) - 2 �����W 4 , ! 13 ) -%$� �=�� " �� , ) - * 13 2 � �� � 4 , ) - *713 � �� , ) - ! 13 2 ���� � 4 , ) - ! 13 $ � � � � �!6 � , ) - * � � � �

Page 9: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

�����������

� ��

�������� ����� ���

��� � ���

�����

� ���

! � " �$# �

% %'&�(

)�*,+ - (/. )10 *,+

232 46535467

46798;:

4=<

4 8

�:âºî>ò{èÁç � "�ÿ-òyàFñ�ç[ìºìºã?æVßÃê ñ�íVè�è�ç[ãÁåtí>ßÃêyâºß{î � õ´ê�å{è�ç[ã�ãÁò{èÁç�êyâäã²àÁè�âºï{òyàÁâºí>ß�æ�ìºíVßÃîgà�á{çIìºâºß{ç�ñníVß{ßÃç[ñ�à�âºß{îàÁá{ç�ñnçnìºì�õ´ñnçnß,àÁç[è�ã U )V æ�ßtê U � � ) V�Ïá{ç=å{èÁçeãÁãÁò{è�çFî>è�æ>êyâºçnß,à�ã$æ�è�ç�çnâ�àÁá{ç[è?çnëÊæ�ìºòÃæ�àÁç[ê�âºß�öÃòÃâºêpå{áÃæ>ã²ç �� �?íVè ��� �$ò{ß{âäþ,ò{ç[ì�ðEêyç �Ãß{çeêpï¹ðàÁá{ç`ëÊæ�ìºò{çÏíVô]à�á{ç`ìºçnë>çnì{ãÁç�à-æ�àRàÁá{ç`âºß,àÁç[è²ô�æ>ñ�ç�÷�Þ¤ß=àÁá{ç7æVïÃãÁçnßÃñnç`í�ôæ�ß=âºó�ó�çnè9ã²çeê(ïtí>ò{ßÃê{æVèÁð � ã²à�æVßÃê{æ�è9êæ�å{å{è�í��yâºóIæÊàÁâºí>ßÃã^í�ô:àÁá{ç(î>è�æ>êyâºçnß,à�ã � ç�÷ î{÷

2 ����BW 4 , *713 ) - � ��, * " ) - � ��, ) -= Wì�çeæVêIàÁí�àÁá{ç?ô�æVó�âºì�âäæVè �ÃëVç?å]íVâºß,à7ã¤à�çnßÃñnâ�ìTô¾í>èÏàÁá{ç �:æ�å{ìäæ>ñ�ç�õ´íVå]çnè9æÊà�íVè[÷!/> C@? E+C�S�S$A�E+CCB�E+IED QPCT?BAGF QRA > I�? Q9HIH CFE+SDCJD QP?BADCFEHGI�J.C ãÁâ�àÁòÃæ�àÁâºíVß é`â�àÁá�æ�ß~âºß,àÁçnèÁô�æVñnçïtçnà¤é-ç[çnßpàÁá{çIñnçnìºìäã U ) V æ�ßÃê U � � ) V âäã?ã²áÃíÊé`ßpâºß �^âºîVòÃèÁç � � à�íVå÷ �ñnñníVè9êyâºß{îIà�íBà�á{çIæVãÁð¹ó�åyàÁíVàÁâäñæ�ßÃæVì�ðyãÁâäãâºß/à�á{çÏìºæVèÁî>ç-êyç[ßÃãÁâ�à¤ð(è9æÊà�âºí»ìºâºó�â�à�é-çÏéÏæ�ß,à�à�í?â�ßtñ�íVè�å]íVè9æÊàÁç àÁá{ç � ò{ó�åtã^â�ßFàÁáÃç�åÃèÁçeãÁãÁò{è�ç�æVßÃêàÁá{ç$å{è�ç[ã�ã²ò{è�ç7î>è�æ>êyâºçnß,à âºßIíVòÃè�ã�ñ9á{çnó�çV÷�(ßÃíÊé`â�ßÃî(àÁá{çeã²ç�à¤é�íFþ,òÃæ�ß,à�â�à�âºç[ãRé�ç$æVèÁç$æ�ï{ìºç»àÁíFñníVßÃã²àÁè�òÃñ�àì�âºß{çeæ�èRå{è�ç[ã�ãÁò{èÁç7ê{âºã²àÁè�âºï{òyà�â�í>ßÃã^íVß=à�á{ç$ì�çnôäà-æ�ßtê�èÁâºî>á>à ãÁâäêyç7íVôàÁá{ç�â�ß,à�çnèÁô�æVñ�ç � ãÁçnç �^âºîVòÃèÁç ��� ï]í�àÁàÁíVó&÷»ã�ã²ò{ó�âºß{î�à�áÃæÊà�àÁá{ç=ñ�ç[ß>à�çnè�í�ô ñ�ç[ìºì � U ) V �7âºã»âºß&öÃò{âäê �� �»æVßÃê&àÁáÃçFñ�ç[ìºì�ñ�ç[ß,àÁçnè�í�ô ñnçnìºì � U � � ) V �`âºßöÃò{âäê � � � � é�ç(ñnæ�ß�é`è�â�à�ç"

Page 10: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

� ! � �M, � � � �� W GJNOK � = W� * � � , * " � � � �� W GJI�K � � � � � = W� � � � � � � * � � !7 7 �� � �� W 8 8 � �� GiILK � � GJI�K� W � �� GJNOK � � GJNOK� W

* � � � �

�Ïá{ç[ãÁç�æ�è�ç/ô¾í>ò{è(ç[þ,òÃæ�àÁâºíVßÃã�ô¾í>è?àÁá{ç�ò{ßÃý,ßÃíÊé`ßÃã � * � � ! ����������� � æVßÃê �������

�� � é`á{âäñ9á ñnæ�ß�ï]çIã²í>ì�ë>ç[ê

ç[æVãÁâºìºð�÷ �ÏáÃç � ò{ó�åtã � � � � � æ�ßÃê 7 7 �� � �� W 8 8�æ�è�çgî>âºëVçnß6ï,ð çeþ>òtæÊàÁâºí>ß � � �/æVßÃê � � � � èÁçeã²å]ç[ñ�àÁâºëVç[ì�ðV÷ �ÏáÃçèÁçeã²ò{ì�à9ã ô¾í>èÏàÁá{ç(åÃèÁçeãÁãÁò{è�ç�î>è�æ>êyâºçnß,à�ã�æ�è�ç"

� � GiNOK� W � � GJNOK� = W Z ��, * " ) - � ��, ) - � � � � � � � � � � � � � GJI�K 7 7 �� � �� W 8 8 [

� � GJI�K� W � � GJILK� = W Z ��, * " ) - � ��, ) - � � � � � � � � � GJNOK 7 7 �� � �� W 8 8 [ � ��� �

é`â�à�á � � � � GJNOK � � � � � � � GJI�K9÷ �Ïðjñ9áÃæ�ß{î>âºß{îIà�á{ç�ã²âºîVßtã»âºßpô¾èÁí>ß>à?í�ô-àÁá{ç � � � � � àÁç[èÁóIã�é�ç�î>ç�à$àÁá{çô¾íVè�óFò{ìäæ>ã7ô¾í>è(àÁá{çBñnæVãÁç�é`â�àÁá å]íVâºß,à U )V âºß�öÃò{âäê ��� �/æ�ßÃêLU � � )V âºß�öÃò{âäê �� ��÷&Þ¦ô �� �(ê{çnß{íVàÁç[ãæ�î>æVâºß�à�á{çIìºâºîVá,à?öÃò{âäê é-çgîVç�à/ô¾íVè � GJNOK � � æ �nç[èÁíEî>è�æ>êyâºçnß,à?ô¾íVè�à�á{çgìºâºîVá,à?öÃò{âäê æ�ßtê�àÁá{ç � ò{ó�åâ�ß à�á{çgåÃèÁçeãÁãÁò{è�ç�îVè9æVêyâºçnß,à�êyâäã�æ�å{å]ç[æVè�ã?ô¾è�íVó+àÁáÃçIô¾íVè�ó�ò{ìäægô¾í>è �������

�� ÷jÞ¤ß~à�á{âäã/ñnæ>ã²çIàÁá{çBå{èÁçeãÁãÁò{è�ç

� ò{ó�å � � � � � âºß ��������� ñ�íVè�è�ç[ãÁåtí>ßÃê{ã?à�íjàÁáÃç $âºèÁâäñ9á{ìºç�à(ï]íVò{ßtê{æ�è�ð�ñníVßÃê{â�à�âºíVß�ô¾íVèFàÁá{çBá{ç[æPë¹ðpöÃò{âäê âºß

àÁá{ç=ìºç[æ>êyâºß{îIíVè9êyç[è$æ>ã²ð¹ó�åyà�í�àÁâäñ(çeþ,òÃæÊà�â�í>ßÃãn÷ �Ãí>è � � � í>è � � � à�á{ç=ç[þ,òÃæÊà�âºíVßÃã �!� � ��èÁçeêyòÃñnç/à�íã²çeñ�íVßtê&íVè9êyçnè»ñnçnß,àÁè9æ�ì�êyâ��]ç[èÁç[ßÃñ�ç/ã²àÁç[ßÃñ�âºìäã�ô¾íVè»àÁá{çFå{è�ç[ã�ã²ò{è�ç(îVè9æVê{â�ç[ß,à7í�ô^öÃò{âäê&å{áÃæ>ã²ç ��� �»íVè7öÃò{âäê�� ��÷ $çnå]çnßÃêyâºß{îpíVß à�á{çgìºíyñnæ�àÁâºíVß�í�ô`à�á{çBñnçnìºì�âºß,àÁçnèÁô�æVñnç U � " )V âºß öÃòÃâºê �� �/íVè/öÃò{âäê ��� �(é�çòÃã²ç�ç[â�àÁá{ç[è W ����� W � í>è W �����W � ô¾íVè�àÁá{çgæVå{å{è�í��yâ�óIæ�àÁâºíVß&íVô�à�á{ç�å{è�ç[ã�ã²òÃèÁç=îVè9æVê{â�ç[ß,à?âºß�ç[þ,òÃæ�àÁâºíVß �!��� �9÷ü¹âºßÃñ�ç`æ�ìºìÃþ,òÃæVß,àÁâ�àÁâºç[ã � ç��{ñnçnåyàRà�á{ç7å{è�ç[ã�ã²òÃèÁç`ëÊæ�ìºò{çeã �M, ) - æ�ßÃê �M, * " ) - â�à9ã²ç[ì�ô � âºß � ��� ��æ�è�ç�ý¹ß{íÊé`ß � òÃãÁâºß{îàÁá{ç=îVè9æVê{â�ç[ß,à�ã�îVâºëVç[ßjâºßpçeþ>òtæÊàÁâºí>ß �!� � �$êyí&ß{í�à�êyâäã²àÁò{è�ïjà�á{çIã²à�æ�ßtê{æ�è9ê �Ãë>ç=åtí>â�ß,à(ã¤à�çnßÃñnâºì^ô¾íVè$àÁá{çêyâäãÁñnèÁçnàÁç �:æ�å{ìäæ>ñ�ç�õ´íVå]çnè9æÊà�íVè[÷ �Ïá{çnðIê{í�í>ß{ìºð=ó�í¹ê{â�ô¾ð�à�á{ç?èÁâºîVá,à átæ�ßÃê�ã²âäêyç�í�ô�àÁá{ç?å{è�ç[ã�ã²òÃèÁç�ùRíVâäãÁãÁíVßç[þ,òÃæÊà�âºíVß �!� � �9÷�Þ¦à»âäãÏâºó�åtí>è²à9æ�ß,àÏàÁí�ýVç[çnå&âºß&ó�â�ßtêgà�áÃæÊà»àÁá{ç/æ>ãÁãÁò{ó�åyà�âºíVßgíVô�æ=ë,âäã�ñ�í>òÃã�öÃò{âäê�îVâºëVçeãòÃã � ò{ó�åpñníVßÃêyâ�à�â�í>ßÃã`ô¾í>è»à�á{ç=å{è�ç[ã�ã²ò{è�ç/î>è�æ>êyâºçnß,à»âºßjï]í�à�á W æ�ßtê � êyâºè�ç[ñ�à�âºíVßÃãn÷"�ÏáÃâºã�çnßtæ�ï{ìºç[ã7àÁá{çå{ò{è�çnìºðjíVßÃç�õ¦êyâºó�çnßÃãÁâºíVßÃæVì)æ�å{åÃèÁí,æVñ9ápô¾íVìºìºíÊé-çeêEá{çnè�ç�÷BÞ¤ß�àÁá{çIâºß¹ë¹âäãÁñnâºê�ñnæVãÁç=é-çIñ[æ�ß~êyç[èÁâºëVç�íVßÃì�ðpæñ�íVßtêyâ�àÁâºíVß�ô¾íVè àÁá{ç$å{è�ç[ã�ã²òÃèÁç � òÃó�åIâºß�àÁá{ç$ß{í>èÁóIæVìÃêyâºèÁçeñ�à�â�í>ß�àÁí�à�á{ç$â�ß,à�çnèÁô�æVñ�çV÷��tò{è²à�á{çnè�ó�íVè�ç � ã²âºßÃñnçàÁá{ç�ëVç[ì�íyñnâ�à¤ðBâäã�ñníVß,àÁâºß¹ò{íVòtã»æVñnèÁí,ãÁã7àÁá{ç=âºß,àÁç[è²ô�æ>ñ�ç � â�à?ãÁçnç[óIã`à�íBï]ç�ô�æ�âºèÁìºðBèÁçeæVãÁíVßÃæVï{ìºç?àÁí�òÃã²ç�íVßÃì�ðíVß{ç7ë>çnìºíyñ�â�à¤ð�æÊà-àÁá{ç�ó�âäêyåtí>âºß>àRíVôTçeæVñ9áIñnçnìºì{ï]íVò{ßÃêÃæ�è�ð�÷^û»í�à�ç»á{íÊé�çnë>çnè^à�áÃæÊà � ò{ó�åÃã�âºß=à�á{ç»ë>çnìºíyñ�â�à¤ðîVè9æVêyâºçnß,à9ã-æVèÁç$ß{í�à»ñníVßÃãÁâäêyçnè�ç[êgá{çnè�ç�÷

�ÏáÃç»îVè9æVê{â�ç[ß,à�ãRî>â�ë>çnß=ï¹ð �!��� ��æVèÁç»ãÁç[ñníVßÃê=í>è�ê{çnè í>ß{ìºðFæ�à àÁáÃç�ó�âäêyå]íVâºß,à�ã^íVôàÁá{ç?ñ�í>èÁè�ç[ãÁåtí>ßÃêyâºß{îç[êyî>ç$âºß&çnâ�à�á{çnèÏöÃòÃâºêBå{áÃæ>ã²çV÷�ü¹âºßÃñnç(é-ç�æVå{å{ìºð�àÁáÃç[ãÁç?ô¾íVè�óFò{ìäæ>ã âºßBàÁá{ç�ó�âäêyå]íVâºß,à-ï]ç�à¤é�çnç[ß&àÁá{ç�ñnçnìºì�õñ�çnß,à�çnè9ã � ç�÷ î{÷ � U � " )V � � é-çFç��yå]ç[ñ�à?ìºíyñnæ�ìºìºð �Ãè9ã¤à$íVè9êyçnè�í�ô æ>ñnñ�òÃè�æ>ñ�ð&æÊà»à�á{ç�â�ß,à�çnèÁô�æVñ�çV÷ �»íÊé�çnë>çnè �ñ�íVß¹ë>çnè�îVçnßtñ�ç(ã¤à�òÃêyâºç[ã7â�ßjãÁç[ñ�àÁâºíVß � ãÁá{íÊé�ìºíyñnæ�ìºìºðIæ�ßEí>è�ê{çnè7íVô æVñnñnò{è9æVñ�ðBæ�è�íVò{ßÃê � ÷ � æ�ßÃê&î>ì�í>ïÃæ�ìºìºðã²çeñ�íVßtêgí>è�êyç[èn÷! E+I�? S@K�CFEHS�C K�CFN�O�J+QRADQPCTS �Ïçeã²âäêyçeã)à�á{ç`âºß,àÁçnèÁô�æVñnç-ë>çnìºíyñ�â�àÁâºçeã�ß{í>èÁóIæVì>à�í?àÁá{ç`âºß,àÁç[è²ô�æ>ñ�çeã�é�çÏáÃæPëVç�à�íæ�å{å{è�í��yâºóIæÊàÁç àÁá{ç7àÁè9æ�ßÃãÁëVç[è�ãÁçÏëVçnìºíyñ�â�à�â�çeã�æ>ãRé�çnìºìh÷)ú ç»êyí?àÁáÃâºã á{çnè�ç`ï¹ð�ãÁâºó�å{ìºç�æPë>çnè9æ�î>â�ßÃî7ëVç[ì�íyñnâ�à�âºç[ãíVßBàÁá{ç�æ>ê � æVñnçnß,à`ç[êyî>ç[ã � ç�÷ î{÷

- , * 13 ) - � �� � - , ) - *713 � - , * " ) - *713 � - , ) - ! 13 � - , * " ) - ! 13 � *

Page 11: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

�Ïá{ç?èÁçeã²ò{ì�à�â�ßÃî/âºß,àÁçnè�å]íVìäæÊà�ç[ê�ë>çnìºíyñ�â�àÁâºç[ã-ã¤à�âºì�ìã�æÊà�âºã²ô¾ðIæ�êyâäã�ñ�è�ç�àÁç?êyâºëVç[èÁî>çnßÃñnç7ô¾è�çnç(ñ�í>ßÃêyâ�àÁâºí>ß ", , * " ) - ! 13 � , , ) - ! 13= W � - , * 13 ) - �$- , * 13 ) - !#"=�� � �.*;�<�=�<�� � CFNPN�� JFC�?BA�CFEHCJD K�C�NPO�J+QPA@QPCTS $â�ë>çnßgàÁá{ç(âºß,àÁç[è²ô�æ>ñ�ç$ëVçnìºíyñ�â�à�â�çeã�é�ç(æ�å{å{è�í��yâºóIæÊàÁç7àÁá{ç�ñníVß¹ëVçeñ�àÁâºë>ç»æ�ßÃê�ë¹âäãÁñníVòÃã àÁç[èÁóIãÏâºßBç[þ,òÃæÊà�âºíVß� �� �?é`â�àÁá~ãÁç[ñ�í>ßÃê íVè9êyç[è(ô¾íVè�óFòÃìºæ>ã$é`á{âäñ9á ñ[æ�ß~ï]çIô¾í>ò{ßÃê âºß àÁç �,à�ïtí¹í>ý¹ã/ìºâ�ý>çFùRç[ð¹èÁçnà/æ�ßÃê �:æPð¹ìºíVè� � ��� ÷��tíVèÏà�á{ç�ñ�í>ß,ë>ç[ñ�àÁâºëVç»àÁçnè�óIã�é-ç?áÃæPë>ç

P � �c� � � � U (.* 13, ) - /���0 , (.*213, ) - , (+* 13, * 13 ) - � , (+* 13, ! 13 ) -= W � - (+*213, ) - , (+* 13, ) - * 13 � , (+* 13, ) - ! 13=��, (.* 13, ) - - (+*213, *713 ) - � - (+*213, ! 13 ) -= W � - (+* 13, ) - - (+* 13, ) - * 13 �$- (+*213, ) - ! 13=�� 4����5

é`â�à�á , (+* 13, ) - �� � , (+* 13, * 13 ) - � , (.* 13, ! 13 ) -� æVßÃê - (+* 13, ) - �

� � - (+* 13, ) - * 13 � - (+* 13, ) - ! 13 *�Ïá{ç?ë,âäã�ñ�í>òÃã-àÁçnè�óIãÏæ�è�ç(ñ�íVó�åÃòyàÁçeêBæ>ã

� = � � (+* 13, ) - ��

/���0 , (+* 13, * 13 ) - � � , (+* 13, ! 13 ) - � , (+* 13, !�3 ) -= W � , (+* 13, * �3 ) - � � , (+* 13, * 13 ) - � , (.*213, ! 13 ) -= W- (+* 13, ) - * 13 � � - (.* 13, ) - ! 13 � - (+* 13, ) - ! �3=�� � - (+* 13, ) - * �3 � � - (.* 13, ) - * 13 � - (+* 13, ) - ! 13=�� 4����5 *

û»í�à�âäñ�çÏàÁátæÊà-é�ç7ß{ç[îVìºç[ñ�à á{ç[èÁç»æ�åtí,ãÁãÁâºï{ìºç � òÃó�åIí�ôëVç[ìºí¹ñnâ�à¤ð�îVè9æVêyâºç[ß>à9ãRæVñnèÁí,ãÁã^à�á{ç»âºß,àÁç[è²ô�æ>ñ�çV÷)Þ¤ßÃñnì�ò{õã²âºíVß�í�ô^à�á{âäã`ç��Tç[ñ�à�é`âºìºìïtç�åtæ�èÁà»í�ô�æ�ô¾íVèÁàÁátñ�íVó�âºß{î�á{âºîVáÃçnè7í>è�êyç[è`âºó�å{ìºçnó�çnß,à9æÊàÁâºí>ß�í�ô^àÁá{ç/å{èÁçeã²ç[ß,àæ�å{å{è�í>æ>ñ9á÷ �Ïá{çÏâºßÃñníVó�å{è�ç[ã�ã²âºï{âºìºâ�à¦ð»ñ�í>ßÃã¤à�è�æVâºß>à^æ�å{åÃì�âºçeê?àÁí?à�á{ç7ñ�ç[ì�ìyñnçnß,àÁç[èÁçeê/ë>çnìºíyñ�â�àÁâºç[ã)æÊà ��� � (+* "ì�çeæVê{ã�à�ígæ�ùRí>âºã�ãÁíVßgçeþ,òÃæÊà�â�í>ßgô¾í>è7à�á{ç�å{è�ç[ã�ã²òÃèÁç � (+* 13 ìºâºë¹â�ßÃîFâºß�àÁá{ç/ß{íyêyç[ã»í�ô^àÁá{çFñ�íVó�åÃòyà�æ�àÁâºíVßÃæVìîVè�âºê÷�ú ç�êyç��tß{ç?àÁá{ç�ê{âºã�ñ�è�ç�à�ç$ê{â�ë>çnè�îVç[ßÃñ�ç�íVå]çnè9æÊàÁí>è`æVñ�à�âºß{îFí>ß�ñnçnìºì�õ´ñnçnß,àÁç[èÁçeê�ëVç[ìºí¹ñnâ�àÁâºç[ã-æVã

� � �B� � , * 13 ) - ! 13 � ��, , ) - � , , ) - ! " � , , !#" ) - � , , ! " ) - ! "= W

� �� -., ) - � -., ! " ) - � -., ) - ! " � , , !#" ) - !#"=�� *�Ïá{âäã�êyâºëVçnè�îVç[ßÃñ�ç�íVå]çnè9æÊà�íVèIñníVè�èÁçeã²å]íVßÃêÃã�àÁí æ�ß�æVå{å{è�í��yâ�óIæ�àÁâºíVß íVô(æ~ê{â�ë>çnè�îVç[ßÃñ�ç&ñ�í>ßÃã¤à�è�æVâºß>à=âºßâ�ß,à�çnîVè9æ�ìtô¾íVè�ó é`â�àÁá&æ=ñ�í>ß>à�èÁí>ìTëVíVìºò{ó�ç�é`â�àÁá�ç[êyî>ç[ã = W æ�ßÃê =�� æVèÁí>ò{ßÃêBß{íyêyç U � " )V � " �ãÁâºß{îàÁá{ç�ãÁæVó�ç�æ�è�îVò{ó�ç[ß,à�ã�æ�ßtê ß{í�à9æÊà�â�í>ß�æVãgïtçnô¾íVè�ç � à�á{ç�ãÁçnó�â$êyâäãÁñnèÁçnàÁçEùRí>âºã�ãÁíVß ç[þ,òÃæÊà�âºíVßAô¾íVègàÁá{çå{èÁçeãÁãÁò{è�ç$æ�àÏàÁá{ç(ß{íyêyçeã`í�ô�àÁá{ç(î>èÁâäêgâºã

�= W " �� , * " ) - ! 13 � � ��BW , * " ) - ! 13 � �� , ) - ! 13 � �����W , ) - ! 13 $� �=�� " �� , *213 ) - � ���� � , * 13 ) - � �� , * 13 ) - ! " � ���� � , * 13 ) - !#" $ � � � �B� 6 � , * 13 ) - ! 13

*�ÏáÃç»å{è�ç[ã�ã²òÃèÁç`î>è�æ>êyâºçnß,à�ãRæVèÁç7ñníVó�å{òyà�ç[ê=òÃãÁâºß{î?àÁáÃç�ã�æ�ó�ç�àÁçeñ9á{ß{âäþ,ò{ç»æVã-ã²áÃíÊé`ß�ï]ç�ô¾í>èÁç»é`â�à�á=àÁá{ç

å{èÁçeãÁãÁò{è�ç?ß{íÊéLìºâºë¹âºß{îFâºß�àÁáÃç�ß{íyêyç[ã7í�ô^àÁá{ç/îVè�âºê÷ 7î,æ�âºß � é�ç(âºßÃñ�í>èÁå]íVè9æÊà�ç�ç �yå{ìºâºñnâ�àÁìºð�àÁá{ç � òÃó�åÃã7âºß

���

Page 12: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

àÁá{ç=å{è�ç[ã�ã²òÃèÁç�æ�ßÃêjàÁá{ç=å{è�ç[ã�ã²òÃèÁçFîVè9æVêyâºçnß,à$â�ßpíVò{è?ã�ñ9á{çnó�çV÷ �(ß{íÊé`âºß{î�à�á{ç=å{èÁçeãÁãÁò{è�ç/âºßjà�á{ç=ß{íyêyç[ãé-ç(ñnæVß�ê{ç��Ãß{ç?à�á{ç(å{è�ç[ã�ã²òÃèÁç$îVè9æVêyâºç[ß>à9ã�ô¾í>èÏàÁá{ç(òÃå]ê{æ�àÁç(íVô�à�á{ç(ëVç[ì�íyñnâ�à�âºç[ã�âºß�ç[þ,òÃæ�àÁâºíVß � �� � "

P ��� U , ) - �/��0 �� 2 � ����BW , ) - ! 13 � � ����BW , ) - * 13 4�� 2 � ���� � , ! 13 ) - � � ���� � , * 13 ) - 4

4���5�tíVè`à�á{ç�ß¹ò{ó�ç[èÁâäñnæVìã²í>ì�ò{àÁâºíVßBí�ô:àÁá{ç�åÃèÁçeãÁãÁò{è�ç$ùRí>âäãÁãÁíVßBç[þ,òÃæÊà�âºíVßÃã�é�ç(òÃã²çFæ�ßEæ�ìºîVç[ï{è9æ�âäñ7ó�ò{ì�àÁâ�õ

îVè�âºêjå{èÁçeñ�í>ßÃêyâ�àÁâºíVßÃç[êEñ�í>ß � ò{î,æÊàÁç�îVè9æVê{â�ç[ß,à�ó�ç�à�á{íyê ������� ÷gø�ßjàÁáÃç �tß{çnè(î>èÁâäê{ã � � � ) ��� � ������� é�çòÃã²òtæ�ìºìºð�æVñ9áÃâ�ç[ëVç�ñníVß¹ëVç[èÁî>çnßÃñnçRâ�ß(ìºçeãÁã�àÁáÃæVß �� â�àÁçnè9æÊà�âºíVßÃãTòÃãÁâºß{î �:õ¦ñ�ðyñ�ìºç[ã)é`â�àÁá � õ � ó�ò{ì�àÁâºî>èÁâäê7ìºç[ëVçnìäãn÷ú�ç�êyâäêBß{í�à7ç[ßÃñ�í>ò{ß,àÁç[è7æ�ß¹ðgñ�í>ß¹ëVçnè�îVç[ßÃñ�ç�å{è�íVï{ìºçnóIã-âºßÃñ�ìºòÃê{â�ßÃîFãÁò{è²ô�æ>ñ�ç$àÁç[ßÃãÁâ�í>ß � ï{òyà7à�á{ç(ß¹ò{ó�ïtç[èí�ô�â�à�çnè9æÊàÁâºí>ßÃã òtã²òÃæVìºì�ð=âºßÃñnèÁçeæVãÁç[ã-é`â�à�áBâºßÃñnèÁçeæVãÁâºß{î/êyç[ßÃã²â�à¤ðIè9æÊà�âºí{÷&&k j � oOtsou��� ��oO~ ou{&|�}�~.yzr�w�Ïá{çBòÃå]ê{æ�àÁç�í�ô�àÁá{ç�ìºçnëVç[ì�õ´ãÁç�à/âäã�æVñ9á{âºç[ëVç[ê é`â�à�á6æpãÁç[ñníVßÃê6íVè9êyçnè�ò{å¹é`âºßÃê ã�ñ9á{çnó�çV÷ Þ¤ß>à�çnî>è�æ�àÁâºß{îç[þ,òÃæÊà�âºíVß � � �-ï]ç�à¤é�çnçnß ���V� ( æ�ßÃê ���V� (.* " é-ç(áÃæPë>ç»à�íIã²çeñ�í>ßÃêgí>è�ê{çnèÏâºßgà�â�ó�ç

] (+* ", ) - � ] (+* ", ) - � = � � �c� � ] � (+*213, ) - � ] (+* ", ) - � = � � � � � � ] � � (+*213, ) - *�Ïá{ç(æVêyë>ç[ñ�à�âºíVßIà�çnè�ó âäã`êyâäãÁñnèÁçnàÁâ��nçeêIæVã

� � � � � ] � � (+*213, ) - , (+* 13, *213 ) - ] (+* 13, * 13 ) - � , (+* 13, ! 13 ) - ] (+* 13, ! 13 ) -= W � - (+* 13, ) - * 13 ] (+* 13, ) - *713 � - (.* 13, ) - ! 13 ] (+* 13, ) - ! 13= � )é`á{çnè�ç , æVßÃê - æVèÁçFàÁáÃç=ê{â�ë>çnè�îVç[ßÃñ�ç/ô¾èÁç[çFâºß,àÁç[è²ô�æ>ñ�ç/ë>çnìºíyñ�â�àÁâºçeã7æÊà?àÁâºó�ç � (+*213 ÷ 7ßtæ�ìºíVî>íVòÃã»àÁíBàÁá{çå{èÁíyñnç[êyò{è�ç�ô¾í>è^àÁá{çÏâºß,àÁç[è²ô�æ>ñ�ç ë>çnìºíyñ�â�àÁâºçeã � àÁá{çÏëÊæ�ìºò{çeã�íVôÃà�á{çÏì�ç[ëVç[ì,ãÁç�à9ãRæÊà:àÁâºó�ç � (+*713 æÊà:àÁá{çÏâºß,àÁçnèÁô�æVñnçæ�è�ç$êyçnàÁç[èÁó�âºß{çeêIï¹ðgæ ��æPð¹ìºíVèÁõ´ç��yåÃæ�ßtã²âºíVß�âºßBãÁåÃæVñnç(æ�ßÃêgàÁâºó�ç�àÁí=ïtíVàÁáEãÁâºê{ç[ã�í�ô:àÁá{ç?âºß,àÁç[è²ô�æ>ñ�ç[ãÏæVßÃêæ�ãÁò{ïÃãÁç[þ,ò{ç[ß,àÏòÃå,é`âºßÃê{â�ßÃî{÷ »ã7æVßBç �yæVó�å{ìºç � é-ç?áÃæPë>ç$ô¾íVèÏàÁáÃç(ì�ç[ëVç[ìTãÁç�à»æ�àÏàÁá{ç(âºß,àÁç[è²ô�æ>ñ�ç7U � " )V

] (+* 13, * 13 ) - ������� �����

] (+*713 ) �, * 13 ) - â�ô , (.* 13, * 13 ) - � �

] (.*213 ) , *713 ) - â�ô , (.* 13, *213 ) - h ��� � ] (.* 13 ) , *713 ) - � ] (.* 13 ) , *713 ) -! çnìäã²ç

é`á{çnè�ç$àÁá{ç(ìºç�ôäà7ãÁâäêyç[êBëÊæ�ìºò{ç$í�ô^]jâäã�îVâºëVç[ßgï¹ð] (.*213 ) �, *713 ) - � ] (, ) - � 2 � ]�BW 4 (, ) - = W� � 2 � ]� � 4 (, ) - = ��� ] (, ) - � 2 � ]�BW 4 (, ) - = W� � " , (, ) - 2 � ]�BW 4 (, ) - � - (, ) - 2 � ]� � 4 (, ) - $ = �� *

�»çnè�ç �u, (, ) - æVßÃê - (, ) - æ�è�ç�àÁá{ç�ý¹ß{íÊé`ß ñ�ç[ì�ì�õ¦ñ�ç[ß>à�çnè�ç[êEë>çnìºíyñ�â�à¤ðEñ�í>ó�åtí>ß{çnß,à�ã?æVßÃêjà�á{çIî>è�æ>êyâºçnß,à�ã$æ�è�çæ�å{å{è�í��yâºóIæÊàÁçeê=òÃãÁâ�ßÃîFò{ßÃì�âºó�â�àÁçeê�ã²çeñ�í>ßÃêgí>è�ê{çnè`îVè9æVê{â�ç[ß,à�ã � âh÷{ç�÷2 � ]�BW 4 , ) - ] , * " ) - � ] , ! " ) -� = W æVßÃê 2 � ]� � 4 , ) - ] , ) - * " � ] , ) - ! "� =�� * � �� �

���

Page 13: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

�ÏáÃç/å]í>ãÁâ�à�âºíVß�í�ô^àÁá{çFâºß>à�çnèÁô�æVñnç(âºã»êyç��ÃßÃç[êjæ>ã`àÁáÃç �nç[èÁí�ìºç[ëVçnì)í�ô�]�÷ �Ïá{ç � õhëÊæ�ìºò{çeã`âºß&ç[þ,òÃæÊà�âºíVß�!��� � æVèÁç$ñnæVìºñnò{ìäæÊà�ç[ê=ò{ßÃê{çnè�àÁá{ç(æ>ãÁãÁò{ó�åyà�â�í>ß�í�ô�æFìºâºß{ç[æVè-êyâäã²àÁè�â�ïÃòyàÁâºíVß=íVôL]Eï]ç�à¤é�çnçnßBà¤é�íFæVê � æVñ�ç[ß,àîVè�âºê�å]íVâºß,à�ãn÷ ��æ�ìºò{ç[ãIí�ô(àÁáÃç�ìºçnë>çnì»ãÁç�à&æÊàIà�á{ç�ß{íyêyçeãIíVô(àÁá{çpîVè�âºê � é`áÃâºñ9á æVèÁçjßÃçnç[ê{ç[ê ô¾íVègàÁá{çå{èÁí � ç[ñ�àÁâºíVß�í�ôTàÁáÃç�ñ�ç[ìºì�õ¦ñ�ç[ß,àÁçnè�ç[êFëVç[ìºí¹ñnâ�àÁâºç[ã � æ�è�ç7çnëÊæ�ìºòÃæ�àÁçeêFï¹ðFà�æVý,âºß{î?à�á{ç»æVèÁâ�àÁáÃó�ç�àÁâäñ�ó�ç[æVß=í�ôTàÁá{çô¾íVò{è7ãÁò{è�èÁí>ò{ßÃêyâºß{î�ñ�çnìºì�õ¦ñ�çnß,à�çnè�ç[ê�ëÊæ�ìºò{çeã�í�ô�]�÷

��íjóIæVâ�ß,à9æ�âºß ] æVã�àÁá{çgãÁâºîVß{çeê�ß{íVè�óIæ�ì�êyâäã²à�æVßÃñ�ç=à�íEàÁá{çIô¾èÁí>ß>à � é-ç�è�çnâºß{â�àÁâäæVì�â��[ça]AæÊôäàÁç[è/çeæVñ9áàÁâºó�ç�ã¤à�çnåpé`â�à�ájæ�ß�æVèÁçeæIñ�í>ßÃãÁçnè�ë,âºß{î�ó�çnàÁá{íyêEêyçeãÁñnèÁâºï]ç[ê&âºß � ��� � �� � òtã²âºß{î=à�á{âºè�êEí>è�ê{çnè��-û?ø ô¾íVèàÁá{ç=ã²åtæÊàÁâäæVì�æ�ßtê�ãÁç[ñ�í>ßÃêEí>è�êyç[è(�»ò{ß{îVçnõ �(ò{à²à�æIô¾íVè$àÁáÃç/à�â�ó�ç�êyâäãÁñnèÁçnàÁâ��[æ�àÁâºíVß�÷ 7ìºì�àÁá{ç�îVç[íVó�ç�à�èÁâäñ[æ�ìâ�ß{ô¾íVè�óIæÊàÁâºí>ßFíVô�àÁá{ç$ô¾è�íVß,à � çV÷ îÃ÷:à�á{ç?ñ�ò{è�ëÊæÊàÁòÃèÁç$æ�ßÃêIàÁá{ç?ß{í>èÁóIæVì{àÁíFàÁáÃç�ô¾è�íVß,à � æVèÁç�ç[æ>ã²âºìºð/í>ïyà�æVâ�ßÃç[êàÁí�ã²çeñ�í>ßÃêgí>è�ê{çnè�ô¾èÁí>ó à�á{ç(ìºçnëVç[ìã²çnà�ãn÷&&k�& �cr�|�w �}�v+��� rsw��yz~(yRr�w��ùRçnè�â�íyêyâäñ�æ�ßÃê�âºß¹ë,âäã�ñ�âäê ��yçeêBéÏæ�ìºìï]íVò{ßtê{æ�è�ðBñníVßÃêyâ�à�â�í>ßÃãÏæ�è�ç?òÃã²çeê�â�ß�à�á{âäã7ã¤à�òÃêyðV÷ ú ç�âºó�å{ìºçnó�çnß,àïtí>ò{ßÃê{æVèÁðEñ�í>ßÃêyâ�àÁâºí>ßÃã`ë¹âäæ=à¤é-íBìäæPðVçnè9ã`íVô îVá{í,ã¤àÁõ´ñnçnìºìäã`æ�è�íVò{ßtê&àÁá{ç�å{á¹ðyã²âäñ[æ�ì�ê{íVóIæ�âºß÷�Þ¤ßEàÁáÃç=ñ[æVãÁçí�ô å]çnè�âºí¹ê{âºñ�ïtí>ò{ßÃê{æVèÁâºç[ã7àÁá{ç�è�ò{ìºç[ã»ô¾íVè#�Ãìºìºâ�ßÃî=àÁá{ç�îVá{í,ã¤àÁõ´ñnçnìºìäã7æ�è�çFã²àÁè9æ�âºî>á>àÁô¾íVè�é�æVè�ê÷��Ãí>è�âºß¹ë¹âäãÁñnâºê��yç[êIéÏæ�ìºìäãRé�ç»ó�âºè�èÁí>èRàÁá{ç$ëVç[ìºí¹ñnâ�à¤ðFæÊà�àÁáÃç�ï]íVò{ßtê{æ�è�ð�æ�ßÃêgç��¹àÁè9æ�å]íVìäæÊà�ç`àÁáÃç�ìºçnë>çnìTã²çnà:]&ë¹âäæ�ìºâºß{çeæ�èç��¹àÁè9æ�å]íVìäæÊà�âºíVß÷&&k�� m$yz�6o � � ~.o�� v ou� ~�v y��~.yzr�wü¹âºßÃñ�ç-íVò{è:ó�ç�à�á{íyê�âäã�æVß�ç��yå{ìºâäñ�â�à)í>ß{ç � é�ç átæPëVç�à�í�è�ç[ã²àÁè�âäñ�à�àÁá{ç-àÁâºó�ç[ã²àÁç[å/ãÁâ��[çRô¾íVèRã²à�æVï{âºì�â�à¤ð�èÁçeæVãÁíVßÃãn÷ú�ç(ìºâºó�â�à�àÁá{ç?à�âºó�ç[ã²àÁçnåBï¹ð

= �^�� � � � ó�âºß ��= ��� ) = ��� ) = � ' �é`â�à�á = � � � ó�âºß, ) - 2 = W, , ) - ) = �- , ) - 4 ��= � � � 2 �����C 4 13 ��= � ' � 2 =��# 4 13æ�ßÃê � � " � � GJNOK � � GiILK �-æ�ßÃê � � ó�â�ß � = WE) =�� �9÷�� �yñnçnåyà�àÁáÃç$ñ[æ�ìäñ�ò{ìäæ�àÁâºíVßÃã:âºßBãÁç[ñ�àÁâºíVß � ÷ �� é`áÃçnè�çé-ç(ãÁç�à � � � �+* ��� é-ç?òÃãÁç[ê�ægÿ � � ß,òÃóFï]çnè`íVô � ÷ �� ÷&&k�� �:|������ r�w �y�~.yzrsw��Þ¤ßBí>è�êyç[è à�í�ñ�íVó�åÃòyàÁç�àÁá{ç?å{è�ç[ã�ãÁò{èÁç�îVè9æVêyâºç[ß>à9ã âºßBç[þ,òÃæÊà�âºíVß � ��� � à�á{ç � ò{ó�åÃã�îVâºëVç[ßIï¹ð � � ��æ�ßÃê � � �óFòÃã²à7ï]ç/çnëÊæVì�òtæÊàÁçeêT÷ �Ïá{çFì�íyñ[æ�ì�ëÊæ�ìºò{çeã`í�ô^à�á{ç/ß{í>èÁóIæVì�ëVçeñ�à�íVè > æ�ßtê&àÁá{ç�ñ�ò{è�ëÊæÊà�ò{èÁç D æ�è�ç�î>â�ë>çnßï,ð > � � ]� � ] � æ�ßÃê D�� � �B>V� ] ��� ] � � � ] � � ] � ] � � ] � � ] �

� ] ��� � ] � � � �3 *�)í�íVïyà9æ�âºß�ëÊæ�ìºò{çeã)æÊà:àÁáÃç ô¾è�íVß,à�� � é�ç ìºâºß{ç[æVèÁìºð7âºß,àÁç[èÁå]íVìäæÊà�ç�ï]ç�à¤é�çnçnßFîVè�âäê?åtí>â�ß,à9ã�÷��tíVè:àÁáÃç�ãÁâ�àÁòÃæ�àÁâºíVßã²á{íÊé`ßBâºß �Ãî>ò{è�ç � é�ç(áÃæPëVç�ô¾íVè`ç �{æ�ó�å{ìºç7ô¾í>èÏàÁá{ç�ñnò{è�ëPæ�àÁò{è�ç"

D � � � � � � � DM, ) - � � DM, * " ) - *�Ïá{çIëÊæ�ìºò{çeã?ô¾íVè > , ) - æ�ßtê D�, ) - æÊà�à�á{çBñnçnìºì�õ´ñnçnß,àÁç[è�ã?æVèÁçIñníVó�å{òyà�ç[ê é`â�àÁá ã²à�æVßÃê{æVè�ê ãÁç[ñ�í>ßÃê íVè9êyç[èêyâ��]ç[èÁç[ßÃñ�ç�í>åtç[è�æ�àÁí>è�ã7æ>ñ�à�â�ßÃî�íVßEà�á{ç�ñnçnìºì�õ´ñnçnß,àÁç[èÁçeêgëÊæVì�òÃç[ã`íVôs])÷�ü¹âºó�âºìäæ�è�ô¾íVè�óFòÃìºæ>ã`æ�å{åÃì�ðBô¾íVè»àÁá{çã²çeñ�íVßtê�å{è�í � ç[ñ�àÁâºíVß � ß{íÊé�é`â�à�áIëÊæ�ìºò{ç[ã-í�ô)àÁá{ç?ìºçnë>çnìtãÁç�àÏìºâºë,âºß{î(âºß�à�á{ç?ß{íyêyç[ã�íVô�àÁá{ç$îVè�âäê � é`á{âäñ9ágæ�è�çæ�ìºé�æPðyã-ñ�íVó�åÃòyàÁçeêgï¹ðgãÁç[ñníVßÃêBíVè9êyçnè`âºß,àÁç[èÁå]íVìäæ�àÁâºíVß�ô¾è�íVó à�á{ç�ñ�ç[ìºì�õ¦ñ�ç[ß,àÁçnè9ê=ëÊæ�ìºò{çeã�÷

���

Page 14: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

�ÏáÃç � ò{ó�åBâºßBàÁá{ç(ßÃíVè�óIæ�ìã¤à�èÁçeãÁãÁç[ã >�� � � ! � � �B> ô¾íVèÏàÁáÃç(â�ß,à�çnèÁô�æVñ�ç?ãÁá{íÊé`ßBâºß �ÃîVòÃèÁç � âäã> � � � � ! � � �B> � �V> � � � 2 GJNOK 4 , ) - �c2 GJI�K 4 , * " ) - � *

ú�ç�ñ�íVó�åÃòyàÁçgàÁá{ç�êyç�ô¾í>èÁóIæ�àÁâºíVßpàÁç[ßÃã²í>èFé`â�à�á~àÁá{ç�êyçnè�âºëÊæÊàÁâºë>ç[ã?îVâºë>çnß~âºß~ç[þ,òÃæ�àÁâºíVß � �� ��÷ ü¹âºó�âºìäæ�èç��yå{è�ç[ã�ã²âºíVßtã�á{í>ìºêgô¾íVè7æVß¹ðIí�à�á{çnè`âºß,àÁç[è²ô�æ>ñ�ç�àÁíVå]íVìºí>îVð�÷

� � ��:Û���×��� kRl � ~.}�~(y �Vvxr�|�w���� |���� �zo6ã²à�æÊà�âäñRèÁí>ò{ßÃê(ïÃò{ï{ï{ìºç âäã�æ�ß�âºß,àÁç[èÁçeã¤à�â�ßÃîÏàÁç[ã²à:ñ[æVãÁç�ô¾í>è:æVß,ð$ß¹ò{ó�çnè�âäñnæVìÊó�ç�àÁáÃí¹ê(ô¾íVè�à¤é-í�å{áÃæVãÁçRöÃíÊé?÷7ì�àÁá{í>ò{îVá/ç��¹àÁè�çnó�ç[ì�ð(ãÁâºó�å{ìºçRâ�ß/àÁá{ç[íVè�ð � à�á{âäã�àÁç[ã²à�ñnæ>ã²ç�îVâºëVçeã�âºó�å]íVèÁà�æ�ß,à:â�ßtã²âºîVá,à)âºß/àÁá{çÏå{è�íyêyòÃñ�à�âºíVßí�ô{à�á{ç`ã²í$ñnæ�ìºìºç[ê(ãÁå{ò{è�âºíVòÃã�íVè:åÃæVè�æ>ã²â�àÁç-ñ�ò{è�èÁç[ß,à�ã � ����� � é`á{âäñ9áFñnæVßFñnèÁçeæÊàÁç-í>ã�ñ�âºìºìäæÊàÁâºí>ßÃãTã¤à�èÁí>ß{î»ç[ß{íVò{î>áàÁí/êyçeã¤à�èÁíÊð/àÁá{ç»â�ß,à�çnèÁô�æVñ�ç ������� ÷^Þ¦à-ãÁá{íVòÃìºê�ï]ç»ß{íVàÁâäñ�çeêFàÁáÃæ�à-ß¹ò{ó�çnè�âäñnæVì,ó�çnàÁá{íyê{ã òÃãÁâ�ßÃî?àÁá{ç»ê{âºã�ñ�è�ç�à�çêyçnì�à�æ�ô¾òÃßÃñ�à�â�í>ßEæ�å{å{è�í>æ>ñ9á � âºßÃñnì�òtêyâºß{î/à�á{ç�ÿÏü�� ó�ç�à�á{íyê{ã � ñnæVß{ß{íVà7èÁç[å{èÁíyêyòtñ�ç?àÁáÃç/ç��{æ>ñ�à»ãÁíVìºòyàÁâºí>ß �çnëVç[ß�â�ô(àÁá{çpîVç[íVó�ç�à�èÁð æ�ßÃê�àÁá{ç�å{è�ç[ã�ã²òÃèÁç � òÃó�åqæ�è�çjîVâºë>çnß ç��{æ>ñ�àÁìºðV÷ �Ïá{çpèÁçeæVãÁíVß âäãIà�áÃæÊàBàÁá{çå{èÁçeãÁãÁò{è�ç�êyâäãÁñníVß,àÁâºß¹ò{â�à¤ðEâäã�æ�å{åÃèÁí��yâºóIæÊà�ç[êEï¹ð æjã²ó�í¹íVàÁá�ô¾ò{ßÃñ�àÁâºíVß é`á{ç[èÁç�à�á{çIå{è�ç[ã�ãÁò{èÁç�î>è�æ>êyâºçnß,àâºã�ß{íVßyõ �nç[èÁíEâºßÃê{òÃñ�âºß{îjæEë>çnìºíyñ�â�à¤ð �Ãç[ìäêT÷ ø�ò{è�ó�çnàÁá{íyê ñníVó�å{ò{àÁç[ã(àÁá{çBç��{æVñ�àFã²í>ìºòyàÁâºíVß é`â�àÁá �[çnè�íå{èÁçeãÁãÁò{è�ç�îVè9æVê{â�ç[ß,à»æ�ßÃê �nçnè�íIë>çnìºíyñ�â�àÁâºç[ãÏåÃèÁíÊë¹âäêyç[ê�à�á{ç�ç��{æVñ�à�å{è�ç[ã�ã²òÃèÁç � òÃó�åjâäã»òÃãÁç[êjâ�ß�ç[þ,òÃæÊà�âºíVß�!� � �9÷ �:âºî>ò{èÁç � ã²á{íÊé7ã�å{è�ç[ã�ã²òÃèÁç=êyâäã¤à�èÁâºï{ò{àÁâºíVßEí>ïyà�æVâ�ßÃç[êjé`â�à�á�íVòÃè$ó�çnàÁá{íyêT÷Iÿ-ìºç[æ�è�ìºð�ãÁçnç[ßpâäã»àÁá{çã²áÃæVèÁågèÁçeã²í>ì�ò{àÁâºíVß=í�ôà�á{ç(êyâäãÁñníVß,àÁâºß¹ò{â�à¤ð�é`â�àÁáÃâ�ßgíVß{ç�îVè�âºêgñ�çnìºìTæ�ßÃê�àÁá{ç �nç[èÁíFå{è�ç[ã�ã²ò{è�ç7î>è�æ>êyâºçnß,à�ã â�ß{õæ�ßÃê/íVòyà9ã²âäêyç àÁá{çÏï{ò{ïÃï{ìºç�÷ �Ïá{ç-è�ç[ãÁò{ì�à�ã�í�ôTêyâ��]ç[èÁç[ß,à^ñníVß¹ëVç[èÁî>çnßÃñnç ã¤à�òÃêyâºç[ã)ô¾íVè:à�á{ç`ã²å{òÃèÁâºíVòtã�ñnò{èÁè�çnß,à9ã

�^âºîVò{è�ç � "�ù è�ç[ã�ã²òÃèÁç$êyâäã¤à�èÁâºï{òyà�âºíVßgæ�ôäàÁçnè`í>ß{ç$àÁâºó�ç[ã²àÁç[å÷æ�è�ç$åÃèÁçeã²ç[ß>à�ç[êBâºß �:æ�ï{ìºçeã ����� � æ�ßÃê � ÷ �ÏáÃç�êyâäæ�ó�ç�à�çnèÏí�ô^ï{ò{ïÃï{ìºç$âäã � � � ñ�ó âºß&æ=ñ�íVó�åÃòyà�æ�àÁâºíVßÃæVìêyíVóIæVâ�ß � � � � � � � ñnó�÷&û�íjè�çnâºß{â�àÁâäæ�ìºâ��[æ�àÁâ�íVß&í�ôÏà�á{çgìºçnë>çnì�ãÁç�à/átæVã�ï]çnç[ß~åtç[è²ô¾í>èÁó�çeêT÷&ùRçnè�â�íyêyâäñïtí>ò{ßÃê{æVèÁðIñníVßÃêyâ�à�â�í>ßÃã æVèÁç$òÃãÁç[êBâºßgç[æ>ñ9á�ê{â�è�ç[ñ�àÁâºíVß�÷ �Ïá{ç(ãÁò{è²ô�æ>ñ�ç�àÁç[ßÃã²âºí>ßBñ�í¹ç���ñnâºçnß,àÏñníVè�èÁçeã²å]íVßtê{ãàÁíFàÁá{ç(ãÁò{è²ô�æ>ñ�ç�àÁç[ßÃã²âºí>ßIïtçnà¤é-ç[çnß�é�æ�àÁç[è�æVßÃêBæVâºè � C � �.* �������

÷ �Ïá{ç?ïtí>ò{ßÃê{æVèÁð�ß{í>èÁóIã� , " ) � æ�è�ççnëÊæ�ìºòÃæ�àÁç[ê&âºßEà�á{ç=æ�è�ç[æIæVèÁí>ò{ßÃê&à�á{çFâºß,àÁç[è²ô�æ>ñ�ç/ê{ç��Ãß{çeêjï¹ð � ] � , ) -� = W ÷ �:æ�ï{ìºç � ã²á{íÊé7ã$êyâ��Tçnè�çnß,àß{íVè�óIã$ô¾íVè/àÁá{çBâ�ß,à�çnèÁô�æVñ�ç�ë>çnìºíyñ�â�àÁâºçeã$æ�ôäàÁç[è/íVßÃç�àÁâºó�ç[ã²àÁç[å�ô¾íVèFæpêyçnßtã²â�à¤ð�è9æÊà�âºí�íVô � GiILK� � GJNOK � � ÷

ü¹âºßÃñ�ç&é�çjã¤à9æ�èÁàIé`â�à�á æVßAâ�ßÃâ�à�âäæ�ìºìºð �[çnè�í�ë>çnìºíyñ�â�à¤ð �Ãçnìäê � à�á{çEâºß>à�çnèÁô�æVñnç�ë>çnìºíyñ�â�àÁâºçeã/æÊôäà�çnè�à�á{ç �Ãè9ã²à

���

Page 15: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

������� � ������� � ����� � ����� � � ���������� � Â��������n¡Á½ ��� � � Â��������n¡Á½ � � � �eÂ��������[¡�½ � � � �  � ��� � ¡�½ ��� ��!� � �!� �  �!��� � ¡Á½ � � �P � � �Ê �������n¡Á½ ��� �Ê � � �P ���!" �[¡�½ ��� �Ê ��� �eÂ#��� " �n¡�½ � � �eÂ�������$�%��� �  �!���!�n¡Á½ � � �PÂ���� �Ê "&� � " ¡Á½ � � �ÊÂ�� " �  �������n¡�½ ��� �Ê ��� � Â������ � ¡�½ �!� �e � �

� ��� ��� ��� �P �����!�n¡Á½ ��� �P � � �Ê " �����[¡Á½ ��� �Ê �!� � Â��������n¡�½ � � �Ê � � �e � � � �[¡�½ � � �e � ���� ��� ��� � �P � " � � ¡Á½ � � �P � � �ÊÂ�� ����� ¡Á½ ��� �Ê � � �  ���!� � ¡�½ ��� �ÊÂ�� � � Â#� � � � ¡�½ � � �e ���

��æVï{ìºç � "�û»í>èÁóIã/í�ô7à�á{ç&â�ß,à�çnèÁô�æVñ�çBëVç[ì�íyñnâ�à¤ð�æÊôäà�çnè � à�âºó�ç[ã²àÁçnå6é`â�à�á�ñníVß¹ëVç[èÁî>çnßÃñnçIè�æ�àÁçeã('&ô¾í>è�æêyçnßÃãÁâ�à¤ðgè9æÊà�â�í � GJNOK � � GJI�K � � ÷ �Ïá{ç/ïtí>ò{ßÃê{æVèÁð�ß{íVè�óIã , " ) � æ�è�ç(çnëÊæ�ìºòÃæ�àÁç[ê�âºß�àÁá{ç�æ�è�ç[æ�æVèÁí>ò{ßÃêàÁá{ç(âºß,àÁç[è²ô�æ>ñ�ç?êyç��ÃßÃç[êBï,ð � ] � = W ÷àÁâºó�ç[ã²àÁç[å æVèÁçEæ êyâºè�ç[ñ�à=ó�çeæVãÁò{è�çBô¾í>è�àÁáÃçjçnè�èÁí>è�âºß�à�á{çEà�á{çjå{è�ç[ã�ã²òÃèÁç&îVè9æVê{â�ç[ß,à�ãn÷�ú�á{çnè�ç[æ>ã�àÁá{çñ�íVß¹ë>çnè�îVçnßtñ�ç�è�æ�àÁç � âºß�àÁá{ç��*)�ëPæVèÁðFïtçnà¤é-ç[çnß � ÷ � õ � ÷ �� é�ç7ã²ç[ç7ñ�í>ß,ë>çnè�îVç[ßÃñ�ç�è�æ�àÁçeã�æ�è�íVò{ßtê � õ � ÷ �â�ßIàÁá{ç � " æ�ßÃê � ßÃíVè�ó&÷!�Ïç[ãÁâäêyç[ã�àÁá{ç$îVìºí>ïÃæ�ì � õRæ�ßÃê � õhßÃíVè�óIã � ��æVï{ìºç � ãÁá{íÊé7ã àÁá{ç$ëVç[ì�íyñnâ�à¤ðFçnè�èÁí>èæÊà7à�á{ç/âºß,àÁç[è²ô�æ>ñ�ç�âºß&àÁáÃç , " ) � ßÃíVè�ó&÷ �»çnè�ç � é�ç/áÃæPë>ç�êyç��ÃßÃç[ê , " ) � � � ��+-,/.,10 " , � é`áÃçnè�ç+ âäã�àÁá{ç=ß¹ò{ó�ïtç[è$íVô âºß,àÁçnèÁô�æVñnç[ã�âºßjàÁáÃç=æVèÁçeæ � ] � = W ÷ �Ïá{ç�ñníVß¹ëVç[èÁî>çnßÃñnç�è�æ�àÁç�ëÊæ�è�â�çeã»ï]ç�à¤é�çnç[ß� ÷ � õ � ÷ � ÷Bü¹âºó�âºìºæVè»ñ�í>ß¹ëVçnè�îVç[ßÃñ�ç�è9æÊàÁçeã(æ�è�ç=íVïyà9æ�âºß{ç[êjô¾íVè/æ&êyç[ßÃãÁâ�à¤ð�è�æ�àÁâºí � GJI�K� � GJNOK � ����� ��� æVãñnæ�ß�ï]çIã²ç[çnß�â�ß �:æ�ï{ìºç � ÷�ÿ-íVó�åtæ�è9æ�ï{ìºç�è�ç[ãÁò{ì�à�ã?æ�ìäãÁígá{í>ìäêjô¾í>è?àÁá{çIñnçnìºì�õ´ñnçnß,àÁç[èÁçeê&ëVçnìºíyñ�â�à�â�çeã»æ�ôäàÁçnèíVß{ç$àÁâºó�çeã¤à�çnå÷

, ) � , " ' , ' , " ) � '���%2 ��� � ÷ � ����� çnõ ��� 3 � ÷ ��� � çnõ ��� 3 � ÷ ��� �� çnõ ��� 3 � ÷ ��� ��� çnõ ��� 3� �(2 � � � ÷ ��� � çnõ ��� � ÷ ��� � ÷ �� �� � çnõ ��� � ÷ ��� � ÷ ������� çnõ ��� � ÷ � � ÷ �� � çnõ ��� � ÷ �����42 �� � ÷ � � � çnõ ��� � ÷ ��� � ÷ ��� ��� çnõ ��� � ÷ �� � ÷ ������ çnõ ��� � ÷ ��� � ÷ ���� çnõ ��� � ÷ ���� � �%2 ��� � � ÷ � � � çnõ ��� � ÷ ��� � ÷ ������ çnõ � � � ÷ ��� � ÷ �� ���� çnõ � � � ÷ � � ÷ ������ çnõ ��� � ÷ �������%2 ���� � ÷ � ��� � çnõ ��� � ÷ ��� � ÷ � �� � çnõ � � ÷ � � ÷ ��� �� çnõ � � ÷ � � � ÷ � ���� çnõ � � � ÷ �

��æVï{ìºç � "�û»í>èÁóIã/í�ô7à�á{ç&â�ß,à�çnèÁô�æVñ�çBëVç[ì�íyñnâ�à¤ð�æÊôäà�çnè � à�âºó�ç[ã²àÁçnå6é`â�à�á�ñníVß¹ëVç[èÁî>çnßÃñnçIè�æ�àÁçeã('&ô¾í>è�æêyçnßÃãÁâ�à¤ð è9æÊàÁâºí � GJNOK� � GJI�K � ������� ÷ �ÏáÃçjïtí>ò{ßÃê{æVèÁð�ß{íVè�óIã , " ) � æVèÁçEç[ëÊæ�ìºòÃæÊà�ç[ê âºß àÁá{ç�æVèÁçeææ�è�íVò{ßÃêIàÁá{ç(âºß,àÁç[è²ô�æ>ñ�ç?êyç��tß{ç[ê�ï¹ð � ] � = W ÷

�:æ�ïÃì�ç � êyâäã²å{ìäæPðyã:ß{íVè�ó íVôTàÁá{ç$ñ�çnìºì�õ¦ñ�çnß,à�çnè�ç[ê�ë>çnìºíyñ�â�àÁâºçeã�æ�à �^� �.*�� ã � ñníVè�èÁçeã²å]íVßtêyâºß{î»à�í � à�â�ó�çnõã¤à�çnåÃã:íVß/àÁá{ç`ñní>æ�è9ãÁç[ã²à�îVè�âäê�æ�ßtê � ��� í>ß�à�á{ç ����52 ���� î>èÁâäêT÷ �Ïá{ç�ñníVß¹ëVç[èÁî>çnßÃñnç�è9æÊàÁçeã:ëPæVèÁð(ï]ç�à¤é�çnç[ß� * � æ�ßÃê �.* � â�ß&à�á{ç�â�ß�Ãß{â�à¤ðgß{í>èÁó æ�ßtêjæ�è�ç�ß{çeæ�è�ì�ðBâäêyçnß,àÁâäñ[æ�ìâºß&àÁá{ç �5) � � " � æVßÃê � ß{í>èÁóIãn÷`Þ¤ßàÁá{ç�ïtí>ò{ßÃê{æVèÁð&ß{í>èÁó é�ç�ãÁçnç�ñ�í>ß,ë>çnè�îVç[ßÃñ�ç�è9æÊà�ç[ã$æ�è�íVò{ßtê � ÷ �Ïá{ç�ò{ß{ç �¹å]ç[ñ�àÁçeêjá{âºîVápñ�íVß¹ë>çnè�îVçnßtñ�çè�æ�àÁç[ã�âºß/à�á{çÏâºß��Ãß{â�à¤ð?ß{íVè�ó æ�ßtê�à�á{çÏïtí>ò{ßÃê{æVèÁð(ß{í>èÁó æVèÁçÏêyò{ç�à�í�à�á{ç`ã²å]ç[ñnâäæ�ìyñ9áÃæ�è9æVñ�àÁç[è�íVô{àÁá{âäã�àÁçeã¤àñnæVãÁç"�ãÁâºßÃñ�ç$àÁáÃç(ç��{æVñ�à»ã²í>ì�ò{àÁâºíVßIâäã`æ �nç[èÁí=å{è�ç[ã�ãÁò{èÁç$îVè9æVê{â�ç[ß,à�âºßÃãÁâäêyç$æVßÃê�íVòyà9ã²âäêyç$à�á{ç(ï{ò{ï{ïÃì�ç � àÁá{çì�íyñ[æÊàÁâºí>ßIí�ô�àÁá{ç(âºß,àÁç[è²ô�æ>ñ�ç$èÁç[ìºæ�àÁâºëVç»àÁí�àÁáÃç�îVè�âäêIâäã�ß{í�à7âºó�å]íVèÁà�æ�ß,à[÷ �Ïá{ç(æ>ñnñ�òÃè�æ>ñ�ð�í�ô:àÁáÃç(å{èÁçeãÁãÁò{è�ç� ò{ó�åjæ�ìºíVßÃç?ê{ç�àÁç[èÁó�âºß{çeã�àÁá{ç�ã²í>ìºòyàÁâºíVß�÷-Þ¦ôRé�ç�òÃãÁç(àÁá{çFç��{æVñ�à»å{è�ç[ã�ã²òÃèÁç � ò{ó�åjæ>ã»æVßEâ�ßÃå{òyà7ô¾í>è»íVòÃèß,òÃó�çnè�âºñ[æ�ì,ó�ç�à�á{íyê � àÁáÃç`èÁçeã²ò{ì�àRâäã:àÁá{ç`ç �{æVñ�à�ãÁíVìºòyà�â�í>ß � òÃå=àÁí(è�íVò{ßÃê{í �Içnè�è�íVè9ã��)âºßÃêyçnå]çnßtêyçnß,à�í�ôtàÁá{ç� õhëÊæVì�òÃç[ãÏêyç�à�çnè�ó�âºß{âºß{î�à�á{ç�âºß,àÁçnèÁô�æVñnç�ìºíyñnæ�àÁâºíVß�÷:ùRí>å{âºß{ç�à7æ�ßtê76æ�ìºç[ãÁý¹âtè�çnå]íVèÁà7âºß � ����� ñ�íVß¹ë>çnè�îVçnßtñ�çè�æ�àÁç[ãRæVèÁí>ò{ßÃê � ÷ �� é`â�àÁá�àÁáÃçnâºè �Ãß{â�àÁçnõhë>íVìºò{ó�ç-ó�ç�àÁáÃí¹ê�ñ�í>óFïÃâ�ßÃç[ê�é`â�à�á=æ$ô¾è�íVß,àÁõNà�è�æ>ñ9ý,âºß{î$æ�ìºîVí>èÁâ�à�á{óô¾íVè`à�á{ç�ç[þ,òÃæVì�êyç[ßÃãÁâ�à¤ðBñnæVãÁç�âºßÃñnì�òtêyâºß{îFë¹âäã�ñ�í>ãÁâ�à¤ð�÷���íÊé-ç[ëVçnè � à�á{çnð&êyí�ß{íVà»ß{í�à�å{èÁíÊë¹âäêyç?è�ç[ãÁò{ì�à�ã�ô¾íVèàÁá{ç(ó�í>èÁç�â�ß,à�çnè�ç[ã²àÁâºß{îFñ[æVãÁç?í�ôRæFáÃâ�î>áBêyç[ßÃãÁâ�à¤ðgè�æ�àÁâºí{÷

� �

Page 16: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

, ) ' , " ' , ' , " ) � '���%2 ��� � ÷ ����� çnõ ��� 3 � ÷ ��� ��� çnõ ��� 3 � ÷ ������� çnõ ��� 3 � ÷ � � � � çnõ ��� 3� �(2 � � � ÷ ������ çnõ ��� � ÷ � � � ÷ ������� çnõ ��� � ÷ ��� � ÷ ����� � çnõ ��� � ÷ � � � ÷ � ����� çnõ ��� � ÷ ���42 �� � ÷ ���� � çnõ ��� � ÷ � � � ÷ ������ çnõ ��� � ÷ �� � ÷ � � ��� çnõ ��� � ÷ ��� � ÷ ��� �� çnõ ��� � ÷ � �� � �%2 ��� � � ÷ ������� çnõ ��� � ÷ ��� � ÷ � ���� çnõ ��� � ÷ �� � ÷ ����� çnõ ��� � ÷ �� � ÷ ������ çnõ ��� � ÷ �����%2 ���� � ÷ ��� ��� çnõ ��� � ÷ ��� � ÷ ����� � çnõ ��� � ÷ ��� � ÷ ������� çnõ ��� � ÷ �� � ÷ ����� çnõ ��� � ÷ � �

��æVï{ìºç � ":û»íVè�óIã�íVô{àÁáÃçÏñnçnìºì�õ´ñnçnß,àÁç[èÁçeê$ëVçnìºíyñ�â�à¤ð$æÊà �^� �.*�� ã�é`â�àÁá/ñníVß¹ëVç[èÁî>çnßÃñnç�è�æ�àÁçeã '7ô¾íVè^æ�êyç[ßÃãÁâ�à¤ðè�æ�àÁâºí � GiNOK� � GJILK � ������� ÷#�Ïá{çFïtí>ò{ßÃê{æVèÁð&ß{í>èÁóIã , " ) � æVèÁçFçnëÊæ�ìºòÃæ�àÁç[ê&âºß&à�á{ç�æVèÁçeæIæ�è�íVòÃßÃê�àÁá{çâ�ß,à�çnèÁô�æVñ�ç?êyç �Ãß{ç[êBï¹ð � ] � � = W ÷� k j � }T��� ou� � |�vx�M} � o��c}OtsoÞ¤ß~àÁá{âäã�ã²çeñ�à�â�í>ß~é-çBå{è�ç[ãÁçnß,à�àÁáÃç�ñníVó�å{òyà9æÊà�â�í>ßÃã(íVô»æ�ãÁò{è²ô�æ>ñ�çIéÏæPëVçgí�ô`à¤é�ípãÁò{åtç[èÁå]í>ãÁç[ê ë,âäã�ñ�í>òÃãöÃò{âäê{ã»æVßÃêjñníVó�åÃæVèÁç$àÁáÃçFè�ç[ãÁò{ì�à�ã`é`â�à�á&àÁá{ç�âºß{â�àÁâäæ�ìëÊæ�ìºò{ç�àÁá{ç[íVè�ðBíVô�ù è�í>ãÁåtç[èÁçnà²àÁâ � ����� ÷ �Ïá{çFâ�ßÃâ�à�âäæ�ìã²ò{èÁô�æVñnç$ê{âºãÁå{ìäæVñnçnó�ç[ß>à�âäã-î>âºëVçnßBï¹ð

� � WE) � � � � + ñ�í,ã ��� W �é`á{çnè�ç � âºã7àÁá{ç�éÏæPëVçnß¹ò{ó�ï]çnè[÷#�Ïá{ç�âºß{â�àÁâäæ�ì�ëÊæVì�òÃçFãÁíVìºòyàÁâºí>ßBô¾í>è»à¤é�íIötò{âäê{ã»é`â�à�á�ç[þ,òÃæVì�ý¹âºß{ç[óIæÊàÁâäñë,âäã�ñ�í,ã²â�à¤ð 2 ñ[æ�ß�ïtç(é`è�â�à²à�çnß&æVã � ����

� � � � � � � � � � � � 2 ���� � � � � � � 2 ��� � + � + ç[è²ô�ñ � 2 � � � 13

� �� , 0 "� , , 2� + � +

� , �c2 � 4 ç �yå � � � , �c2 � � � ��çnèÁô�ñ � � ,�� 1 � � )é`á{çnè�ç$àÁá{ç � , ã`æ�è�ç$àÁá{ç?ô¾í>ò{è`è�í,íVà�ãÏíVô)àÁáÃç�æ�ìºîVç[ï{è�æVâäñ`ç[þ,òÃæ�àÁâºíVß

� � � � � ��� 2 � 13 � � � � � � � � � � � 2 � � � � � � � � � ��� 2 � � 3 � � � � � � � � 2 � � � + � �+*�Ïá{ç� , ã=æ�è�çgîVâºë>çnß~ï¹ð� " � � � � � " � � � � � � � " � � � � � � " ��é`â�àÁá�ñ�ðyñ�ìºâäñ�å]çnè�óFòyà9æÊà�â�í>ßÃã(íVô7àÁá{çâ�ßtêyâäñ�ç[ã-ô¾í>è� ) � ) � ÷ �Ïá{ç�âºß¹ë,âäã�ñ�âäê�í,ãÁñnâ�ìºìäæÊà�âºíVß�ô¾èÁçeþ,ò{çnßÃñnðIâäã + � C � � � � GJNOK � � GJI�K � � æVßÃêBàÁá{çêyâºó�çnßÃãÁâºíVß{ìºç[ã�ãRåÃæVè�æVó�ç�à�çnè � âäãÏêyç �Ãß{ç[ê&æ>ã � � � GJNOK � GiILK� � � GJNOK � � GJI�K � ÷�ÏáÃç?ñníVó�å{ò{à�æÊà�âºíVßÃã-é-ç[èÁç(ñ[æ�è�èÁâºç[ê=í>òyà`ô¾íVèÏà�á{ç?é�æPë>çnß¹ò{ó�ïtç[è � � � é`â�à�á�æVßBâºß{â�àÁâäæVìtæ�ó�åÃì�â�à�òÃêyç� + � �+* � � � � íVßEæ�ãÁþ,òÃæVèÁç?î>èÁâäêBé`â�à�á � � � � � � ��� ÷ �Ïá{âäã`æ�ó�å{ìºâ�àÁòtêyç$ñníVè�èÁçeã²å]íVßtê{ã-àÁí � îVè�âäêBñnçnìºìäãíVß�íVò{è �Ãß{çeã¤à � ����� 2 ����� �´õ´ó�ç[ãÁá÷�ú�ç(òÃãÁç?åtç[èÁâºíyêyâäñ7ï]íVòÃßÃê{æ�è�ðIñ�í>ßÃêyâ�àÁâºí>ßÃã âºßBàÁá{ç?á{í>èÁâ��[íVß,à�æVì]æVßÃêâ�ß¹ë¹âäãÁñnâäê ��yç[êBé�æVì�ìñníVßÃêyâ�à�â�í>ßÃã�âºßIàÁáÃç(ëVçnèÁàÁâäñ[æ�ìêyâºèÁçeñ�à�â�í>ß÷

�:æ�ïÃì�ç � ãÁá{íÊé7ã-àÁá{ç(è�ç[ãÁò{ì�à�ã�í�ôRæ=ñ�í>ß,ë>çnè�îVç[ßÃñ�ç$ã¤à�òÃêyðIô¾í>èÏàÁá{ç(çeþ,òÃæ�ì�êyç[ßÃãÁâ�à¤ðIñ[æVãÁç�÷^ú ç(å{èÁçeã²ç[ß,àñ�íVß¹ë>çnè�îVçnßtñ�ç�è�æ�àÁç[ãRô¾íVèRà�á{ç»ñ�ç[ìºì�õ¦ñ�ç[ß,àÁçnè�ç[ê/å{è�ç[ã�ã²ò{è�ç � (+* 13, ) - � àÁá{ç»ñnçnìºì�õ´ñnçnß,àÁç[èÁçeê�ë>çnìºíyñ�â�à¤ð , (, ) - � æVßÃêFàÁá{çñ�çnìºì�õ¦ñ�çnß,à�çnè�ç[ê=ìºçnë>çnìã²çnà7ëÊæ�ìºò{ç[ã:] (, ) - ÷ �Ïá{ç(ñ�í>ß,ë>çnè�îVç[ßÃñ�ç$è�æ�àÁç�'�ô¾í>è7æ�ß¹ðIëÊæ�è�âäæ�ï{ìºç�dAâäã`êyç �Ãß{çeê�æ>ã

' � ìºí>î � ���é`â�à�á � � � d ��� � d � �

� d � � d � � *�»çnè�ç � � � = W�� =�� âäãÏà�á{ç�î>èÁâäê�ã²åÃæ>ñ�âºß{î=æ�ßÃê�d ( � êyçnßÃí�àÁçeãÏàÁáÃç/ã²í>ìºòyàÁâºíVßBíVß�àÁáÃç�îVè�âäêgé`â�àÁá&î>èÁâäêã²åÃæ>ñ�âºß{î � � ÷�Þ¤ßFà�á{ç � õ�æ�ßtê � õhß{í>èÁó é�çÏáÃæPëVçÏñ�í>ß¹ëVçnè�îVç[ßÃñ�ç è�æ�àÁç[ã^æVèÁí>ò{ßÃê � âºß/çeæVñ9á�ëÊæ�è�âºæVï{ìºç�íVß/àÁá{ç

��

Page 17: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

î>èÁâäê{ã ëÊæ�è�âäæ�ï{ìºç ' � � ��� ' � � � 1 ' � � � 3� �(2 � � � �� 2 �� � �����(2 ����� , � ÷ �� � ÷ ��� � ÷ ��� � ÷ ��� � ÷ ��� � ÷ ���] � ÷ ��� � ÷ � � ÷ ���� 2 �� � ��� �%2 ����� � �����%2 ���� , � ÷ �� � ÷ ��� � ÷ ���� � ÷ � � � ÷ �� � ÷ ���] � ÷ �� � ÷ ��� � ÷ ��

��æVï{ìºç � "�ÿ-íVß¹ëVç[èÁî>çnßÃñnç�è9æÊà�ç[ã-ô¾íVè`à�á{ç(å{è�ç[ã�ã²òÃèÁç � à�á{ç�ë>çnìºíyñ�â�à¤ð � æ�ßÃêgàÁáÃç�ìºçnëVç[ìã²çnà`ô¾ò{ßÃñ�à�âºíVßBô¾íVè`à¤é�íîVè�âºêpã²çeþ,ò{çnßÃñnç[ã(âºß~êyâ��Tçnè�çnß,à(ß{íVè�óIã(æÊà � � ����� ãn÷ � GJNOK � � GiILK � ������� ý¹î �Êó � � 2 GJNOK � 2 GJILK ���� õ � ó ��ã � C�� �.* ��� � û��ÊóI÷�Ãß{çnè`î>èÁâäêIãÁç[þ,ò{ç[ßÃñ�çV÷�»ã`ç �¹å]ç[ñ�àÁçeê � àÁáÃç?ñníVß¹ëVç[èÁî>çnßÃñnç»è9æÊà�ç�ô¾í>èÏàÁá{ç?å{è�ç[ã�ãÁò{èÁç$âºßgà�á{ç?âºß��Ãß{â�à¤ð,õhßÃíVè�óâºãÏï]çnìºíÊéqà¤é-í � ï{òyà$é`â�à�á&ëPæVìºò{ç[ãÏíVô ' � � * ��� æVßÃê ' � � * � � ïtçnà²à�çnè7à�áÃæ�ß&à�á{ç�ç �¹å]ç[ñ�àÁçeê �Ãè�ã²à7íVè9êyç[èæVñnñnò{è9æVñ�ð=âºßBàÁá{âäãÏß{í>èÁó&÷ �Ãí>èÏàÁá{ç(ë>çnìºíyñ�â�à¤ð=é-ç?áÃæPë>ç?ãÁç[ñníVßÃêBíVè9êyç[è�íVô^æ>ñnñnò{è�æ>ñ�ð=âºßBçeæVñ9á�ß{íVè�ó&÷

»ì�àÁá{í>ò{îVáAé�ç�ñnæVß{ß{í�àIç �yåtçeñ�àBæ~åÃèÁâºíVè�ð ñ�í>ß¹ëVçnè�îVç[ßÃñ�ç�à�íÊé�æVè�ê{ã�à�á{ç�æVßÃæ�ìºð,àÁâäñ�ã²í>ì�ò{àÁâºíVß � â�àIâäãâ�ß,à�çnè�ç[ã²àÁâºß{î�à�í�ñ�íVó�åtæ�è�ç�é`â�àÁá�â�àn÷ �:æ�ïÃì�ç � ãÁá{íÊé7ãÏæ�ñ�íVß¹ë>çnè�îVçnßtñ�ç�ã²àÁòtêyðgíVô�à�á{ç(âºß>à�çnèÁô�æVñnç�ìºíyñnæ�àÁâºíVßàÁíÊéÏæ�è9ê{ãÏàÁáÃç=æVßÃæ�ìºð,àÁâäñ(ãÁíVìºòyà�â�í>ß÷#�ÏáÃç/çnè�è�íVè � � � � � ����� âäã»êyç �Ãß{çeê�æVã�à�á{çFè�í¹í�à$ó�ç[æ�ßjã�þ,òÃæ�è�ç/íVôRàÁá{çæ�ó�å{ìºâ�àÁòÃê{ç?ê{â �Tçnè�çnßtñ�ç/ï]ç�à¤é�çnç[ßEàÁáÃç/ìºâºß{ç[æVè`àÁá{ç[íVè�ðEã²í>ì�ò{àÁâºíVß�� 1 , ( ��� � � � � � � + � � ñ�í,ã � � �7æ�ßtê&àÁá{çñ�íVó�åÃòyàÁçeêIâºß,àÁçnèÁô�æVñnç�ìºíyñnæ�àÁâºíVßBæÊà� � � "

� � � 1 , ( ��� � � �� `/� + � � 1 , ( ��� � � � � � � � � � )f� � � � ��� 13 *�à�á{âºîVáÃçnè»î>èÁâäê&è�ç[ãÁíVìºòyà�â�í>ßÃãÏé�çFáÃæPë>ç/ãÁò{ï{ìºâºß{ç[æVè»ñ�í>ß,ë>çnè�îVç[ßÃñ�ç?à�íIàÁáÃçFìºâºß{ç[æVè7âºß{â�àÁâäæ�ì)ëPæVìºò{ç(àÁáÃçníVè�ð�÷ü¹âºó�â�ìäæVè:è�ç[ãÁò{ì�à�ã�á{íVìäê=ô¾í>è�æFêyçnßÃãÁâ�à¤ðFè9æÊà�âºí � � ������ é`á{âäñ9áIæ�è�ç»êyâäãÁå{ìäæPðVçeêFâºß �:æ�ï{ìºç � ÷��Ïá{ç$ãÁò{ï{ìºâºß{çeæ�èñ�íVß¹ë>çnè�îVçnßtñ�ç�àÁíÊéÏæ�è9ê{ã$àÁáÃçgâºß{â�àÁâäæ�ìRëÊæVì�òÃç=àÁá{ç[íVè�ðpâºßÃêyâäñ[æÊàÁç=à�áÃæÊàFé-ç�æ�è�ç�ß{í�à=ñ�íVß¹ë>çnè�îVâºß{îBàÁíÊéÏæ�è9ê{ãàÁá{âäã ãÁíVìºòyà�âºíVß÷ �ÏáÃç�ñ�í>ó�å{òyà9æÊàÁâºí>ßÃã^í�ô $ò{ç[ð��=ç[è-çnà�æVìN÷ � �� � æ�ßÃê=ùRí>å{âºß{ç�à æVßÃê 6TæVì�çeã²ý¹â � ����� ãÁá{íÊé�ç[êæ�ìäã²í&í>ß{ìºðpãÁò{ï{ìºâºß{ç[æVè�ñ�íVß¹ë>çnè�îVçnßtñ�ç�ô¾íVè/àÁá{âäã?à�ç[ã²à�ñnæ>ã²çV÷ �Ïá{çnð æ�è�îVòÃç[ê�âºß � ����� à�áÃæÊàFàÁá{çgïtí>ò{ßÃêyçeêêyíVóIæVâ�ßgâ�ßBàÁáÃç�ñnæ�ìäñnò{ìäæÊàÁâºí>ßIñ�íVòÃìºêgïtçFæ�ñ[æ�òÃãÁç$ô¾íVèÏà�á{âäã`êyâäãÁñnèÁç[åÃæ�ßÃñnðIæVã�à�á{ç(âºß{â�àÁâäæ�ì]ëÊæ�ìºò{ç?ã²í>ì�ò{àÁâºíVßâºã�î>â�ë>çnßBâ�ßBàÁáÃç(â�ß�Ãß{â�àÁç?êyçnå{àÁáEñnæ>ã²çV÷ ��íÊé-ç[ëVç[è � àÁçeã¤à9ã`é`â�àÁá&êyíVòÃï{ìºâ�ßÃî�àÁáÃç�ñ�íVó�åÃòyà�æ�àÁâºíVßÃæVì]ê{íVóIæ�âºßâ�ßIàÁá{ç$ëVç[è²à�âºñ[æ�ì]êyâºèÁçeñ�àÁâºí>ß=ãÁá{íÊé�ç[êIæVßgâºßyöÃòÃçnßÃñnç»íVßgàÁá{ç?ãÁíVìºòyàÁâºí>ßÃãRï{òyàÏà�á{çnðIêyâäê�ßÃí�à`ñ9áÃæ�ß{î>ç»ó�òÃñ9áàÁá{ç�ñníVß¹ëVç[èÁî>çnßÃñnç7àÁíÊéÏæ�è9ê{ã àÁá{ç�æVßÃæ�ìºð,àÁâäñ�ã²í>ì�ò{àÁâºíVß÷

� � � 1 , ( ��� � '� �(2 � � � ÷ ����� � çnõ � 3��42 �� � ÷ � ��� � çnõ � � ÷ ���� � �%2 ��� � � ÷ ��� �� çnõ � � ÷ � ����%2 ���� � ÷ ������� çnõ � � ÷ �

��æVï{ìºç � "-ÿ-íVß¹ë>çnè�îVçnßtñ�ç7íVô�àÁá{ç$çnè�è�íVè ï]ç�à¤é�çnç[ßIà�á{ç(ñ�íVó�åÃòyàÁçeêIæ�ó�å{ìºâ�àÁòtêyç � � � ��æ�ßÃêIàÁá{ç?æ�ó�åÃì�â�à�òÃêyçí�ôTàÁá{ç»ì�âºß{çeæ�èRâºß{â�àÁâäæVì,ëÊæVì�òÃçÏàÁá{ç[íVè�ð � 1 , ( ��� � ÷ � GJNOK � � GJI�K � ������� ý¹î �Êó � � 2 GJNOK � 2 GJILK � ��� õ � ó ��ã �C � �.* ����� û��ÊóI÷Þ¤ß �:æ�ïÃì�ç � é�çIã²áÃíÊé à�á{çIè�ç[ãÁò{ì�à�ã?í�ô`æ&îVè�âäêpñníVß¹ëVç[èÁî>çnßÃñnç�ã²àÁòtêyð�ô¾í>è/æEêyç[ßÃã²â�à¤ðjè9æÊà�âºí�íVô ����� ÷

�Ïá{ç»ñ�í>ß¹ëVçnè�îVç[ßÃñ�ç`è9æÊà�ç[ã�ê{í/ß{íVà�êyâ��Tçnè ãÁâºîVß{â��tñ[æ�ß,àÁìºð?ô¾è�íVó"àÁá{ç$ñ�í>ß,ë>çnè�îVç[ßÃñ�çÏè9æÊàÁçeã�é`â�àÁá�çeþ,òÃæ�ìÃöÃò{âäêêyçnßÃãÁâ�àÁâºç[ã»âºß ��æVï{ìºç � ÷�ø�ß�àÁá{ç �Ãß{ç[è?îVè�âºêEãÁç[þ,ò{ç[ßÃñ�çFé-ç�áÃæPë>çFãÁç[ñ�í>ßÃêEí>è�êyç[è�ñníVß¹ëVç[èÁî>çnßÃñnç(ô¾íVè�àÁá{ç

���

Page 18: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

� � � 1 , ( ��� � '� �(2 � � � ÷ ������ çnõ ��� 3��42 �� � ÷ � � � çnõ ��� � ÷ ���� � �%2 ��� � � ÷ ����� � çnõ ��� � ÷ �������%2 ���� � ÷ ������ çnõ ��� � ÷ �

��æVï{ìºç � "-ÿ-íVß¹ë>çnè�îVçnßtñ�ç7íVô�àÁá{ç$çnè�è�íVè ï]ç�à¤é�çnç[ßIà�á{ç(ñ�íVó�åÃòyàÁçeêIæ�ó�å{ìºâ�àÁòtêyç � � � ��æ�ßÃêIàÁá{ç?æ�ó�åÃì�â�à�òÃêyçí�ô)à�á{ç(ìºâºß{ç[æVè à�á{çní>èÁð � 1 , ( ��� � ÷� $çnßÃãÁâ�à¤ðIè9æÊàÁâºí � � ������ çeþ>òtæ�ìý¹âºß{çnóIæ�àÁâäñ7ë¹âäãÁñní>ãÁâ�àÁâºç[ãn÷ëVçnìºíyñ�â�à¤ð , æ�ßtê&àÁá{ç�ìºçnë>çnì�ãÁç�à ] âºßjç[æ>ñ9áEß{íVè�ó&÷#�Ïá{ç�ñ�í>ß,ë>çnè�îVç[ßÃñ�ç(è9æÊà�ç[ã`ô¾í>è»à�á{ç�å{èÁçeãÁãÁò{è�ç�âºß&àÁá{ç� õ�æ�ßÃê � õhß{í>èÁó æVèÁç(ãÁìºâ�î>á,àÁìºðFìºçeãÁã-àÁátæ�ß&âºßIàÁáÃç(ç[þ,òÃæ�ì�êyçnßÃãÁâ�à¤ðIñnæ>ã²çV÷

î>èÁâäê{ã ëÊæ�è�âºæVï{ìºç ' � � � � ' � � � 1 ' � � � 3� �(2 � � � �� 2 �� � ��� �%2 ����� , � ÷ � � ÷ ��� � ÷ ���� � ÷ � � � ÷ � � ÷ �� ] � ÷ �� � ÷ � � � ÷ ������ 2 ��� ��� � �%2 ��� � � �����(2 ����� , � ÷ ��� � ÷ ��� � ÷ ���� � ÷ � � � ÷ � � � ÷ ���] � ÷ ��� � ÷ ��� � ÷ ���

��æVï{ìºç � "�ÿ-íVß¹ëVç[èÁî>çnßÃñnç�è9æÊà�ç[ã-ô¾íVè`à�á{ç(å{è�ç[ã�ã²òÃèÁç � à�á{ç�ë>çnìºíyñ�â�à¤ð � æ�ßÃêgàÁáÃç�ìºçnëVç[ìã²çnà`ô¾ò{ßÃñ�à�âºíVßBô¾íVè`à¤é�íîVè�âºê ãÁç[þ,ò{ç[ßÃñ�çeã/âºß6êyâ��Tçnè�çnß,à/ß{í>èÁóIãFæÊà � � ��� � ãn÷ � GJNOK ��� ý,î �Pó � � � GJI�K � ������� ý¹î �Êó � � 2 GJNOK �2 GJI�K � ��� õ � ó �Êã � C � �.* ����� û��ÊóI÷

�^âºîVòÃèÁç � ã²á{íÊé7ã�à�á{ç�àÁâºó�çFç[ëVí>ì�ò{àÁâºíVßjíVô à�á{ç�ã²ò{èÁô�æVñnç�æ�ó�å{ìºâ�àÁòÃê{ç�ô¾íVè$à�á{ç�ç[þ,òÃæVì�êyçnßÃãÁâ�à¤ðEè�æ�àÁâºíñ�íVó�åÃòyà�æ�àÁâºíVß�æ�ßÃê�à�á{ç � � ��� � ê{çnßÃãÁâ�à¤ð�è9æÊàÁâºíEñnæ>ã²çV÷ �ÏáÃçIß¹ò{ó�çnè�âäñnæ�ìRãÁíVìºòyà�â�í>ßpâäã?ß{çeæ�è�ìºðjâºßÃêyâäã¤à�âºßyõîVò{âäãÁáÃæ�ï{ìºç?ô¾è�íVó à�á{ç=â�ßÃâ�à�âäæ�ì�ëÊæ�ìºò{ç=ãÁíVìºòyàÁâºí>ßBô¾í>è?ç[þ,òÃæ�ìRêyç[ßÃãÁâ�à¤ð�öÃòÃâºêÃã�÷(Þ¤ßpàÁá{ç�á{âºîVá�êyçnßtã²â�à¤ð&è�æ�àÁâºíñnæVãÁç/é�ç�íVïÃãÁçnè�ëVç/æ�ãÁâ�î>ß{â��tñnæVß,à�êyâ��]ç[èÁç[ßÃñ�çFâºßEàÁá{ç=å{átæVãÁç�ãÁåtç[ç[ê�ïtçnà¤é-ç[çnßjà�á{ç�æ�ßÃæVìºð>à�âäñnæ�ì�ã²í>ì�ò{àÁâºíVßæ�ßÃê àÁá{çEñnæVìäñ�ò{ìäæÊà�âºíVß÷ »ßyô¾í>è²à�ò{ßÃæÊà�çnìºð � ß{çnâ�àÁáÃçnè�â�ß � ����� ß{í>èFâºß ����� � åÃâºñ�àÁò{è�ç[ã=æVèÁç�ã²á{íÊé`ß�âºß~àÁá{çò{ß{ç[þ,òÃæVì�êyçnßtã²â�à¤ðIñnæ>ã²çV÷��Ïá{ç�ã²áÃâ�ôäà`âºß�é�æPë>ç$ãÁåtç[ç[ê�âºßÃñ�è�ç[æ>ã²çeã é`â�à�áBâºßÃñnèÁçeæVãÁâ�ßÃî/êyçnßtã²â�à¤ðIè9æÊà�â�íÃ÷� k�& � }E�:�RoEyzp�n � m }O� �zr^v yzw � ~(}�� yz�zy�~<� ñ�íVó�ó�í>ßjà�ç[ã²à�å{èÁí>ï{ìºçnó ô¾í>è�ß¹ò{ó�çnè�âäñnæVì�ó�çnàÁá{íyê{ã?ô¾í>è?à¤é-íVõhåÃáÃæVãÁç�öÃíÊé âäã$àÁáÃç �»æPð¹ìºçnâºîVáyõ �:æPð,ìºí>èâ�ßtã¤à9æ�ï{âºìºâ�à¤ð � é`áÃçnè�ç�æpá{ç[æPë¹ð öÃòÃâºê6âäã/å{ìäæ>ñ�ç[ê íVß à�íVå í�ô�æpìºâºîVá,à�öÃò{âäê6é`â�à�á�æ�ã²óIæVìºì âºß{â�àÁâäæ�ì-åtç[è²õàÁò{è�ïÃæÊà�âºíVß�íVô à�á{ç�âºß>à�çnèÁô�æVñnçFï]ç�à¤é�çnç[ßpà�á{ç�à¤é-í�öÃòÃâºêÃã�÷=Þ¤ß�àÁá{ç�ìºâ�àÁç[è�æ�àÁò{è�ç/í>ßpß¹ò{ó�ç[èÁâäñ[æ�ì:ó�ç�àÁáÃí¹êÃãô¾íVè(à¤é�í�õ´å{áÃæ>ã²ç�öÃíÊé"íVß ��yç[ê î>èÁâäê{ã?í>ß{çIñnæVß �ÃßÃê~óIæVß,ðpþ>òtæ�ìºâ�à�æÊà�âºëVç/è�ç[ãÁò{ì�à�ã$ô¾í>è(àÁáÃçIìºíVß{î�àÁâºó�ç �ß{íVßyõ´ìºâºß{ç[æVè�çnë>íVìºòyàÁâºí>ßIí�ô�àÁá{ç?âºßÃã²à�æ�ïÃâ�ìºâ�à¤ð � ã²ç[ç»ô¾í>è`ç��{æ�ó�åÃì�ç � � � � ����� ������ ������� � ������� üyò{èÁåÃèÁâäãÁâ�ßÃîVìºð �ô¾èÁí>ó àÁáÃç�ñ�â�àÁçeêBè�ç�ô¾ç[èÁç[ßÃñ�çeãÏíVß{ìºð��Ïè9æVñ9ý¹ï{âºìºìtçnà»æ�ìh÷ÃñnñníVó�åÃæVèÁçeêgâºß � � àÁáÃçnâºè`èÁçeã²òÃì�à9ã�ô¾íVè»ãÁá{í>è²à`à�â�ó�çeãþ>òtæ�ß,àÁâ�à�æ�àÁâºëVç[ìºð/é`â�àÁáBàÁá{ç(ìºâºß{çeæ�è-àÁá{ç[íVè�ð�÷� }E�:�RoEyzp�n � m }O� �zr^vVyzw � ~.}���yR�zy�~<��� ��n�r^vx~@~.yz�6o�ÃíVè$ã²áÃíVèÁà7àÁâºó�ç[ã»æ�ßjâºß{â�àÁâäæVì�ìºð�ãÁâ�ß¹òÃãÁíVâäê{æVìêyâäã¤à�ò{èÁïtæ�ßÃñnç�îVè�íÊéÏàÁáEæVñ[ñ�í>è�êyâºß{î�à�ígìºâºß{ç[æVèÏàÁá{ç[íVè�ðBç �¹å]í�õß{çnß,àÁâäæVì�ìºð�âºßgà�â�ó�ç?æVãÏç �yå � � � ��é`â�àÁá&æ�îVè�íÊéÏàÁáBè�æ�àÁç?îVâºëVç[ßgï¹ð �������

� � � # /0 � � C# � � GJNOK � � GJI�K 45 ) � ��� �

���

Page 19: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

0 200 400 600 800 1000 1200Time t [s]

−1

−0.5

0

0.5

1

Am

plit

ute

y/y

0

Linear theoryNumerical solution on 256x256 grid

ρ(Α)/ρ(B) = 1

0 500 1000 1500Time t [s]

−1

−0.5

0

0.5

1

Am

plit

ud

e y/

y 0

Linear theoryNumerical solution on 256x256 grid

ρ(B)/ρ(A)

= 100

�:âºî>ò{èÁç � " �-ëVíVìºòyà�âºíVßgíVô^àÁá{ç�êyâºó�çnßÃãÁâºíVß{ìºç[ã�ã æVó�å{ìºâ�àÁòÃêyç � � � ��� � + ô¾í>è7àÁáÃç/âºß{â�àÁâäæ�ìTëÊæ�ìºò{ç�ãÁíVìºòyà�âºíVß�í�ôàÁá{ç=ìºâºß{ç[æVèÁâ��nçeê�ç[þ,òÃæÊà�âºíVßÃã$æ�ßÃê�àÁáÃç�ß,òÃó�çnè�âºñ[æ�ì:ã²í>ì�ò{àÁâºíVß÷ �)ç�ôäà "(ç[þ,òÃæ�ìRêyç[ßÃãÁâ�à¤ðjñ[æVãÁçFé`â�à�á � GJNOK �� GJI�K � ��� ��� ý,î �Pó � � è�âºîVá,à�" � GJNOK � ��� ý,î �Pó � æ�ßÃê � GJI�K � ��� ��� ý¹î �Pó � ÷ �Ïá{ç(ß¹ò{ó�çnè�âäñnæVì]ãÁíVìºòyà�â�í>ßÃãã²á{íÊé`ß�æ�è�ç(ñ�íVó�åÃòyàÁçeêgí>ß�æ �����%2 ���� îVè�âäê�é`â�àÁá 2 GJNOK � 2 GJILK � ��� ! � ó ��ã � CY� �+* ��� � û��Póg÷é`á{çnè�ç � âäã`à�á{ç/éÏæPëVç(ß¹ò{ó�ï]çnè�æVßÃê àÁá{ç �à¤é-í¹íyê&ß¹ò{ó�ïtç[è � � � GiNOK � � GiILK �3��� � GJNOK � � GJI�K ��÷ú�â�à�á � ��� � é�ç?ñ[æ�ß�êyç��ÃßÃç?æ�ñ�è�â�àÁâäñnæVìtãÁò{èÁô�æVñ�ç»àÁç[ßÃã²âºí>ß C � ô¾íVèÏé`á{âäñ9á � � � ÷ $ç��tß{âºß{î/à�á{ç(ã¤à9æ�ï{âºìºâ�à¤ðåÃæ�è9æ�ó�çnàÁçnè

� � CC � )é-ç`áÃæPë>ç`ã¤à9æ�ï{ìºçÏí>ã�ñ�âºìºìäæÊàÁâºí>ßÃãí�ôTà�á{ç7âºß,àÁçnèÁô�æVñnç�ô¾íVè�� � � æVßÃê=âºßÃã¤à9æ�ï{âºìºâ�à¤ð�é`â�à�á�ç �yåtí>ß{çnß,àÁâäæVìyîVè�íÊéÏàÁáô¾íVè�� h � ÷��:âºî>ò{èÁç � ã²áÃíÊé7ãÏíVßBàÁá{ç(ìºç�ôäà»ãÁâäêyç�àÁá{ç?à�â�ó�ç$çnë>íVìºòyà�â�í>ß�ô¾íVè�êyâ��]ç[èÁç[ß>àÏëÊæ�ìºò{çeãÏí�ô)à�á{ç�ã²à�æÊõï{âºì�â�à¤ð(åÃæVè�æVó�ç�à�çnè��»÷ �)íVî,æ�è�â�àÁá{ó�âäñ æVó�å{ìºâ�àÁòÃêyçÏëÊæ�ìºò{çeãRæ�è�ç»ãÁá{íÊé`ß�ô¾íVè�à�á{ç7ò{ßtã¤à9æ�ï{ìºç7ãÁâ�à�òÃæÊà�âºíVßÃãRæVßÃêì�âºß{çeæ�è�ìºðIãÁñ[æ�ìºç[ê&æ�ó�åÃì�â�à�òÃêyç[ã�ô¾í>è7à�á{çFã²à�æVï{ìºç/ñnæ>ã²ç(é`â�àÁá�� � � * ��� ��� ÷ �Ïá{ç/å{è�ç[ãÁçnß,à�ç[êEñnæVìäñ�ò{ìäæÊà�âºíVßÃãé-ç[èÁçIñ[æ�è�èÁâºç[êpíVò{àFé`â�àÁá æVß~â�ßÃâ�à�âäæ�ì�ãÁâ�ß¹òÃãÁíVâäê{æVìRã²ò{èÁô�æVñnçgêyâäã²àÁò{è�ïÃæ�ßtñ�ç�í�ôÏà�á{çIô¾í>èÁó �.* � � � �^ñ�í>ã ��� W �9÷�Ïá{çBñníVó�å{òyà9æÊà�â�í>ßÃæ�ì-êyíVóIæVâ�ß âäã$� � � � � æ�ßÃê � � � � � �Eé`â�à�á ��72 ����� îVè�âäê åtí>â�ß,à9ã�÷�ùRçnè�â�íyêyâäñïtí>ò{ßÃê{æVèÁð�ñ�í>ßÃêyâ�àÁâºí>ßÃã âºß�à�á{ç?á{íVè�â��ní>ß>à9æ�ì]æ�ßÃê ��yç[êgé�æVì�ìTãÁì�âºågñníVßÃê{â�à�âºíVßÃã-æ�è�ç�òÃãÁç[êgâºßIàÁáÃç$ë>çnèÁàÁâäñnæVìêyâºèÁçeñ�àÁâºí>ß÷ �Ïá{çBêyçnßÃãÁâ�à¤ð�íVô�à�á{çIá{çeæPë¹ð�æVßÃê�àÁá{çgì�âºî>á>à?öÃòÃâºê~æVèÁç � GJNOK � ����

�æVßÃê � GiILK � ���

��

èÁçeã²å]ç[ñ�àÁâºëVç[ì�ð � ñ�í>èÁè�ç[ãÁå]íVßÃêyâºß{î$à�íFæ�ß �à¤é-í¹íyê=ß¹ò{óFï]çnè-í�ô � �.*�� ÷��Ïá{ç$ñ�è�â�à�âäñnæ�ìtã²ò{èÁô�æVñnç`àÁç[ßÃãÁâ�í>ß�âäãC � � ������� 3 ÷ �^â�î>ò{è�ç � ãÁá{íÊé7ãÏíVß�à�á{ç(èÁâºî>á>à`ãÁâºê{ç�à�á{ç/êyâäãÁåtç[è�ãÁâºíVßgê{âºæVîVè9æ�ó!é`â�àÁáEñ�í>ó�å{òyà�ç[êBîVè�íÊéÏàÁá

è�æ�àÁç[ãÏë>çnè9ã²òtã�àÁá{ç(î>èÁíÊéÏà�áBè9æÊà�ç[ãÏîVâºë>çnßgï¹ð �!��� �9÷ �ÏáÃç�ß¹ò{ó�çnè�âäñnæ�ì]îVè�íÊéÏàÁá�è9æÊàÁçeã`æ�è�ç(æ�ìºé�æPðyã�æ=ìºâ�àÁàÁìºçì�çeãÁã�àÁáÃæVß à�á{çBìºâºß{çeæ�è(à�á{çní>èÁð ï{òyà�àÁáÃç�æVîVè�çnçnó�ç[ß,à�âºã�þ,ò{â�àÁçBèÁçeæVãÁíVßÃæVï{ìºç�÷jÞ¦à=âäã/þ,ò{â�àÁç�êyâ���ñnò{ì�à/à�íåtç[è²ô¾í>èÁó ñ�í>ó�å{òyà�æ�àÁâºíVßtã$æVèÁí>ò{ßÃê�à�á{çBñnèÁâ�àÁâäñ[æ�ìRã²òÃè²ô�æ>ñ�ç�à�çnßÃãÁâºíVß÷Eû»òÃó�çnè�âºñ[æ�ìRç �¹å]çnè�âºó�çnß,à�ã?é`â�à�á æô¾çné àÁâºó�ç�õ¦ã²àÁçnåtã:ãÁá{íÊé�ç[ê�âºßÃã²à�æ�ïÃâ�ìºâ�à¤ð$ô¾íVè C�� �� .* ����

� 3 æ�ßÃê�ã²à�æVï{âºì�â�à¤ð$ô¾íVè C � � �+* ����� 3 � ñ�í>èÁè�ç[ãÁåtí>ßÃêyâºß{îàÁí�ëÊæVì�òÃç[ã�íVô7àÁáÃçBã²à�æ�ïÃâ�ìºâ�à¤ð�åtæ�è9æ�ó�ç�à�çnè�íVô � � �+* æ�ßÃê�� � � * �� ÷ �ÏáÃçBêyâäãÁåtç[è�ãÁâºíVß~êyâäæ�î>è�æVó

å{èÁíÊë¹âäêyçeê�âºß � � ãÁá{íÊé7ãÏò{ß{åÃá,ðyãÁâäñnæ�ìTâºßÃã²à�æVï{âºì�â�à¤ðFí�ô:àÁá{ç[â�èÏß¹ò{ó�ç[èÁâäñnæVìtó�çnàÁá{íyêBçnë>çnßBô¾íVè�� � � * �� ÷�ÏáÃç�èÁçeã²òÃì�à9ãTí�ô{æ`ñ�í>ß¹ëVçnè�îVç[ßÃñ�ç^ã²àÁòÃêyð$í�ô,àÁáÃç�ç��yå]íVß{ç[ß>à�âäæ�ì�î>èÁíÊéÏà�á�è9æÊà�çRé`â�à�á(æ`ñ�íVßtã¤à9æ�ß,à)ã¤à9æ�ï{âºìºâ�à¤ð

åÃæ�è9æ�ó�çnàÁçnè�� � �.* ��� æVèÁç/îVâºëVç[ßEâ�ß ��æVï{ìºç � ÷ �Ïá{çFàÁá{ç[íVè�ç�à�âºñ[æ�ì�î>èÁíÊéÏà�ájè9æÊàÁçFâäã � 1 , ( ��� � � � * ������ ÷�Ïá{ç�ß¹ò{ó�çnè�âäñnæVìîVè�íÊéÏàÁáEè9æÊàÁçeã»æVèÁç�êyçnàÁç[èÁó�âºß{çeêBï¹ðEæ�ìºâºß{ç[æVè7ñ�ò{è�ëVç �{à�âºß&àÁáÃçFæ�ó�åÃì�â�à�òÃêyç?è�æVß{îVç/í�ô� *�� ìºß � � � � ��� � + � �+* � ÷ �Ïá{çgß,òÃó�çnè�âºñ[æ�ì^îVè�íÊéÏàÁá è9æÊà�ç�íVß�àÁá{ç �Ãß{çeã¤à/î>èÁâäêpâäã � ÷ ��� ïtç[ì�íÊé àÁá{çì�âºß{çeæ�è àÁá{ç[íVè�ð�÷

���

Page 20: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

0 2 4 6 8 10Time t [s]

−1

0

1

2

3

4

Dim

ensi

on

less

am

plit

ud

e

ln(a/a0): Φ=0, 0.125, 0.25, ... , 0.9375a/a0: Φ=1.0625

0 0.2 0.4 0.6 0.8 1Stability Parameter Φ

0

1

2

3

Gro

wth

rat

e n

linear theorycomputed growth rates

������������ �������� ��������������� �"!#$��&% ��'�!��(�)�"*+-,/.10�2435,768�1� �9*+�#:;<�<����=>%?!���<�@�?�A�����B% C�-�<�"��#���D'"%>E�)% ��F�<�HGJILK+M�N�OP��M<O�O<OFM�Q R�OP��M/Q$S�OTR?��M4N+Q OPN �UC�� �)�$��';� ��*+�����-�)��=>% !#�"<�H�?���V�-���)% W���!��(�YX�'"% �)% ��F�<�Z[I\K�]HK�^(Q>N ��]_K�^`N���]&K+^`a�R>�+]_K+^P��]&K+^`b�N �+]AK+^(R>�+]AK+^`S�R?��]AK�^ c�a�R?��]dQ ^`K b7N �+^fe���� V7�� g�����';��4����� �h*+�#E%?� �4%?�jid�(�)VkCF����'��+�$*l%?�"*m�)V���������C�% !n� �)�>i&�Vo�4%>�$�U�1�)� �p!��#�"<%?���)V�����X ^Jq�!�!_CF����'��+�4%>������"�id�(�)Vsr;t(uwv3$r�t(x�vyIza�� �g%{b?|o}zQ>N?S�� �)��*�id�#�VD~��sI�K+^P��~��UIzN5�n^

�4�g�����T�(� �Y�F��� �a7N8}�b | Q ^`b?|�R�Q<�E�K�Q Eb |�}JQ>N?S R�^`c c |�Q<�E�K7N Q ^`K?|Q5N S�}gN��?b a+^PN>|7S c?�E�K7N Q ^`a KN � b�}g�+Q5N c+^(R$|�R$| �E�K�a Q ^(R$|

��%?W�!�US+ H�_� ��=���)� ���CF@� ���)V�UF�+';� ��<�7���%?!/� �)�>i&�V��)%?� � �1� �&r+�53<r��dI�a�]"GJILK�O`N��?G"�

�����&���� Y¡£¢�¤5¥��w�d�Y¦n§L Y¨�© ª+��«� ��Y ¬ª?�n­U�Y¦£¨�¡®ª� �¯°�± ���@!�% ���@������<���C�% !wF��%?��'"!#9���d�)V�8C�� ��'����)%>�)��� �g�?�£e@%5X�!������V+E���%5X�!�� �_���"���)% W���!��(�)��<�A�#�g�)V�²!�����������?]��"� �+E�!��#�"<%?���)�� ���� ^�³msC�� �"���*+��9V�<�{�)V�´��+% ��'�!�²�)�$%>�)<*m����µ N>|�]yN7R<¶·^g�&V�s*+<�"���#�)�#$��?�H�V�´!���� V7�U%?�"*��)V�´V�$%5=7XJ¸"����*o%?�)�K+O#Q$b c | ¹��"3$�´º²% �"*hQ OPN N��>¹���35�´ºF]n�)<�'�$C»�)�#=��!�X?]�idV"�#!�8�V�*+X��"%?����C�=7���)CF�7���#�YX{����$% C4V�¸"����*����H¼�I½a�O(Q5a�¾ Q$K�¿;º»¹��"3$�´�4^&�&V�U¹�������´%?���C@=����)CF����#���<�A%?�)��V�<�À t(uwv IÁQ O`S?|�R RU¾�Q5K+¿ � � � 3>��%?�"* À t(x�v IÂN�OP� � �+QU¾�Q5K+¿/º»� � 3>�»^z�&V�ÃCF����'��+�4%>������"%?!&*+� �´% ���k���~��´I®Q��Ä%?�"*J~��{IÅ|_�Æid�#�)VÃ%DQ5N S�}JN � b�� �)��*/^�³o²�"�²';��)���+*+��CUW������"*�% �XgCF���"*+�#������"�d���g�V�V�� �)��Ç�� �7�4%?!n%?�"*mÈ"��$*oiH% !�!£W������"*�% ���$�9�#�o�)V�´=������C�% !£*+���$C»������w^g�&V"´�����#���%?!y';������)W"%>�)��� �o���� ��= <�´W�X´É�.BÊ;2AILK�O K�Q$~��yCF�7�"Ë+Êgid�#�VDË´ILN>�y35~���^

�y��� �"�-b���V��>i��d�V�8�#�7�)���B% CF8%?�"*g=��!��+CF�#�YX´= $C»�)� �4��%>�9*+�#:;<�<���@�����<��^@q@���V��V�<%5=�Xs¸"����*'�<��F�)�)%?�<�d�V�8!#����V��&¸"�"��*g���&�1����´�d%´�8�"�V��)��� �[�V"%?';<*g<�"*D'�����V������{�)V�-!���� V7�&¸"����*g��'w^@�&V"��������!�%>������Ã��V��>i��U�� ��{'����"C4Vo� :f�?�dV�$%5=7Xø"����*o%>�-�)V����U� �)��*J�)<�� !��+�)�#���w^´�&V�´�)<���!#�)�U% �{���� ���+*�% � �)����<�7�Aid�#�V�'���C»�)���$�H'��$��<�7�<*����mµ N7R<¶·^

ÌY�o�y��� ���)�R´iA��)�';<%?�U�V�´C�� ��'��+�4%>�)�#����% W��>=�8���"C�!#��*+�����������B% C�8�)��"���� �mid�#�Vo%D������B% C���������� �DC��7�Í´CF����7���?�_GlIÎK+O#Q$a�a�R$¹���3>� � ^��&V�²����8��!�%?���� �s*����� ���Y�)�)%?�<�@CF!�<% �!�X{�V"8���)%?W"�#!���Ç������F:/<C»�9�?�_�V�{���"���B%�CF-�)��"���� �gÏg�V"8V�$%5=�X�¸�����*D���$*������ �)U�����²��s';��"F��4%>�)²�)V�{!��#��V7�d¸"����*%?�"*s�V�UV"<%5=�X�¸"����*�'���<C�<�&%>���)��d'����"C4V+E��?:J�V��>iL�)V�9�<�"*+<�"CFXs��{�1� �)�Ð�)� ����*�*+�)� '"!#��)��^

Q$c

Page 21: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

t = 0.2s t = 0.4s t = 0.6s t = 0.7s t = 0.8s t = 0.5s t = 1.0s

�:âºî>ò{èÁç � " �»æPð¹ì�ç[âºîVáyõ ��æPð¹ìºíVè$âºßÃã²à�æ�ïÃâ�ìºâ�à¤ðjíVô$æ�ß�âºß{â�àÁâäæ�ìºìºð�ãÁâ�ß¹òÃãÁíVâäê{æVì�å]çnèÁàÁò{è�ïÃæÊà�âºíVß6æ�àFà�â�ó�çeã �b��.* � ) �+* � ) �+* � ) �.* � ) �.* � ) �.* ) � * � ã�÷^ÿ-í>ó�å{òyà�æ�àÁâºíVßtæ�ìêyíVóIæVâºßIâäã � ó 2 � ó é`â�àÁá �� 2 ����� îVè�âºêBå]íVâºß,à�ãn÷� GJNOK � �+* ��� �� ý¹î �Êó � � � GJI�K � � * � ��� ý¹î �Pó � �+A GJNOK � A GiILK � �+* �� � ��� ! � ý¹î �PóIã� �"ÙFÖEÜ��nÛ��:ÝnÙ/Ö �(ÖEÚ��VÛ�×:Û&Ø����!Ù�Ø��ú�ç7átæPëVç7êyçeãÁñnèÁâºï]ç[ê=æ(ß{ç[é ß,òÃó�çnè�âºñ[æ�ì¹ó�ç�à�á{íyê�àÁí/ãÁíVìºë>ç-à�á{ç»âºßÃñníVó�å{è�ç[ã�ãÁâ�ïÃì�ç û�æPë¹âºçnèÁõ¦ü,à�íVý>ç[ã:ç[þ,òÃæ�õàÁâºíVßÃã�ô¾íVèFà¤é-íVõhåÃáÃæVãÁç�öÃíÊé é`â�àÁá�ã²ò{èÁô�æVñnç�àÁç[ßÃãÁâ�í>ß÷6ø�ò{èFæVå{å{è�í>æVñ9á~âäã/îVò{âäêyçeê~ï¹ð æVß6æVãÁð¹ó�åyàÁíVàÁâäñæ�ßÃæVì�ðyãÁâäã íVô:à�á{ç�çeþ>òtæÊàÁâºí>ßÃãÏâºßBà�á{ç�ìºâºó�â�à-í�ô�æVß�âºß��tß{â�à¤ðBêyç[ßÃã²â�à¤ðIè9æÊà�âºí{÷�ú ç/ç �¹åÃì�âäñnâ�à�ìºðFâºßÃñníVè�åtí>è�æ�àÁçàÁá{ç � ò{ó�åAâºß à�á{çEå{è�ç[ã�ã²ò{è�ç&æ�ßÃê�àÁáÃçEå{èÁçeãÁãÁò{è�çBî>è�æ>êyâºçnß,àFâºß,àÁípàÁáÃçjêyâäãÁñnèÁçnàÁâ��[æ�àÁâºíVß~ã�ñ9á{çnó�çV÷�ø�ò{èæ�å{å{è�í>æ>ñ9á�âäã(çeæVãÁðjàÁíEâºó�å{ìºç[ó�çnß,à�âºß~æ�ß¹ð�åÃèÁí � ç[ñ�à�âºíVßpà¤ð,å]çIó�çnàÁá{íyêpô¾íVè(à¤é�í�õ´å{áÃæ>ã²ç�öÃíÊé?÷�Þ¤ß�àÁá{çå{èÁçeã²ç[ß,àÁç[êpô¾í>èÁó5â�à/ó�íyêyâ��Ãç[ã?í>ß{ìºðjàÁáÃçgè�â�î>á,à?átæ�ßÃê ãÁâäêyç�âºß�àÁá{çBå{èÁçeãÁãÁò{è�ç�ùRíVâäã�ã²í>ß�ç[þ,òÃæ�àÁâºíVß~æVßÃêêyí¹ç[ã`ß{íVà»æ �Tç[ñ�à`àÁá{ç/ã²à�æVßÃê{æVè�ê �Ãß{â�àÁç�ê{â �Tçnè�çnßtñ�ç?íVå]çnè9æÊà�íVè9ã�÷ �Ïá{ç�ñníVó�å{ò{à�æÊà�âºíVßÃæVì]ç �{æ�ó�å{ìºç[ã�î>â�ë>çnßã²á{íÊé àÁáÃæ�à:à�á{ç`ó�ç�à�á{íyê�áÃæ>ãRæ»î>ìºíVïÃæVì,æVå{å{è�í��¹âºóIæ�àÁâºíVß$íVè9êyçnè^íVô � æ�ßÃê/àÁátæÊàRâ�àRñníVß¹ëVç[èÁî>ç[ã�ô¾íVè�ãÁçnëVç[è�æVìàÁç[ã²à7ñnæ>ã²çeã�àÁíÊéÏæ�è9ê{ãÏæ�ßÃæVì�ð,à�âºñ�ã²í>ìºòyàÁâºíVßtã�í�ô^ìºâºß{ç[æVèÁâ��nçeê=å{è�íVï{ìºçnóIãn÷Þ¦à7âäã-àÁç[ó�åyàÁâºß{î�à�í�óIæVýVç�àÁá{ç(ó�çnàÁá{íyêBìºí¹ñ[æ�ìºìºð�æ>ãÏé-ç[ìºìTæ>ãÏîVìºíVïtæ�ìºìºðFãÁç[ñ�í>ßÃêBíVè9êyçnè[÷��Ïá{âäã�ñ[æ�ß�ïtç

æVñ9á{âºçnë>ç[ê�ï¹ð�à�æ�ý¹âºß{îBâ�ß,à�ígæVñ[ñ�íVòÃß>à$á{âºîVá{ç[è7à�áÃæ�ßpì�âºß{çeæ�è7åÃèÁçeãÁãÁò{è�ç/êyâäã¤à�èÁâºï{ò{àÁâºíVßÃãÏâºßjó�â��yç[êEñnçnìºìäã7í>èï,ðFâºß>à�èÁíyêyòtñ�âºß{î$á{âºîVá{ç[è � ò{ó�å�ñ�íVßtêyâ�àÁâºíVßÃã:ô¾í>èRàÁáÃç7å{è�ç[ã�ã²ò{è�ç�÷ �Ïá{âäã^é`âºì�ìyï]ç`à�á{ç7ß{ç �,à�ã¤à�çnå�âºß�ô¾ò{è²à�á{çnèêyçnë>çnìºíVå{ó�ç[ß,à�ã�í�ô$àÁá{âäã�ó�çnàÁá{íyêT÷ »ß íVå]çnß�þ,ò{ç[ã²àÁâºí>ß�ìºç�ôäàBï,ð6àÁá{çpã²âºó�ò{ìäæÊàÁâºí>ßÃã�íVô$à�á{ç�ê{æVó�åtçeêã²ò{èÁô�æVñnç(é�æPë>ç(âºß�ü¹ç[ñ�àÁâºíVß � ÷ � âäã`à�á{ç/ç[èÁè�íVè`âºßEå{átæVãÁç/ã²å]çnçeê&ô¾íVè�á{âºîVáEêyç[ßÃã²â�à¤ð�è�æ�àÁâºí>ãn÷�Þ¦à�é�íVò{ìäê�ïtçâ�ß,à�çnè�ç[ã²àÁâºß{î(à�íFâºß¹ëVçeã¤à�â�î,æÊà�çÏàÁá{âäã-ô¾ò{èÁàÁá{ç[èÏàÁí �ÃßÃê�íVòyà�àÁáÃç?ñ[æ�òÃãÁç�ô¾íVè�àÁá{âäãÏêyâäã�ñ�è�çnåÃæVßÃñ�ð=ï]ç�à¤é�çnçnßgàÁá{çâ�ßÃâ�à�âäæ�ìÃëÊæVì�òÃç?ãÁíVìºòyà�âºíVßIíVô�à�á{ç(ìºâ�ßÃç[æ�è�â��nçeêFû$æPë¹â�ç[è²õ¤ü,à�íVýVçeã çeþ,òÃæÊà�â�í>ßÃã�æ�ßÃêBàÁáÃç(ß,òÃó�çnè�âºñ[æ�ìtó�ç�àÁáÃí¹ê÷

üyígô�æVè�é�ç/áÃæPë>çFæ>ãÁãÁò{ó�ç[êEñníVß,àÁâºß¹ò{â�à¤ðgíVôRàÁá{ç=ë>çnìºíyñ�â�à¤ðIîVè9æVê{â�ç[ß,à�ãn÷#�Ïá{âäã�âºã»íVï¹ë¹âºíVòÃãÁìºðBæIè�íVòÃîVáæ�å{å{è�í��yâºóIæÊàÁâºí>ß=ô¾íVè�öÃò{âäê{ã7é`â�à�ájêyâ��]ç[èÁç[ß>à�ë,âäã�ñ�í,ã²â�àÁâºçeã�÷ �Ïá{ç � ò{ó�åjñníVßÃê{â�à�âºíVß � íVßjàÁá{ç/à�æ�ßÃîVçnß,à�âºæVìã¤à�èÁçeãÁãÁç[ã:å{è�íÊë¹âºê{ç[ã^ã²í>ó�ç`æVê{êyâ�à�â�í>ßÃæ�ì,âºßyô¾íVè�óIæÊà�âºíVß/æVïtí>òyàRàÁá{ç � òÃó�å�â�ßFàÁáÃç7ëVç[ì�íyñnâ�à¤ð?î>è�æ>êyâºçnß,à^æ>ñ�è�í>ã�ãàÁá{ç(âºß,àÁç[è²ô�æ>ñ�ç$é`á{âäñ9áBñ�í>ò{ìäêIï]ç(òÃãÁç[êBàÁí=âºó�å{è�íÊëVç7à�á{ç(ó�ç�à�á{íyêT÷

�^âºßÃæVì�ìºð � à�á{çnè�ç&âäã�æ�ãÁçnëVç[èÁç�ìºâºó�â�à9æÊà�â�í>ß�à�í�ñníVó�å{ò{àÁç�èÁçeæ�ìÏå{á¹ðyã²âäñnæVì-åÃèÁí>ï{ìºçnóIã�é`â�à�á�í>ò{è�à¤é-íVõêyâºó�çnßÃãÁâºíVßÃæVìeó�çnàÁá{íyêT÷�Þ¤ß(àÁáÃç�ô¾òyà�ò{èÁç�â�à)é`âºìºì�ïtç�ß{çeñ�ç[ã�ã�æ�è�ð7àÁí�ç��¹àÁç[ßÃê?à�á{ç ó�çnàÁá{íyê?à�í�æ �yâäãÁð,ó�ó�çnàÁè�âäñå{èÁí>ï{ìºçnóIã�æ�ßÃê�àÁá{è�çnç`ãÁåÃæVñnç�êyâºó�ç[ßÃã²âºí>ßÃã�÷�ø»à�á{çnè^ô¾òyà�ò{è�çÏê{â�è�ç[ñ�àÁâºíVßtã�í�ôÃàÁá{âäã:é-í>èÁý(âºßÃñnìºòÃêyç ç �¹àÁçnßtêyâºß{îàÁá{ç�æ�ìºîVí>èÁâ�àÁáÃó ô¾íVè»àÁá{ç=ñ�í>ó�å{òyà9æÊàÁâºí>ß�íVôRöÃíÊé7ã�é`â�à�ájëÊæ�è�âäæ�ï{ìºç(êyçnßtã²â�à¤ð&æ�ßÃêjãÁò{èÁô�æVñnç�àÁç[ßÃãÁâ�í>ß � á{ç[æ�ààÁè9æ�ßÃã²ô¾çnè�æ�ßÃê�ëÊæVåtí>èÁâ��[æ�àÁâºíVß � æ�ßÃê � ç[ëVçnß,à�òÃæ�ìºìºð � ñ9áÃçnó�âäñnæVìyèÁçeæVñ�à�âºíVßÃãRë¹âäæ?ötæVó�ç`ô¾è�íVß,à�àÁè9æVñ9ý¹âºß{î�æVì�í>ß{î

� �

Page 22: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

t = 0.5s t = 0.7s t = 0.9s t = 1.0s t = 1.1s t = 1.2s t = 1.3s

�:âºî>ò{èÁç � " ��æPð,ìºç[â�î>áyõ ��æPð¹ìºíVèTâºßÃã²à�æ�ïÃâ�ìºâ�à¤ð7é`â�àÁá=ãÁò{è²ô�æ>ñ�ç-àÁç[ßÃã²âºí>ß÷ � ë>çnè�ð,àÁá{âºß{î�âºã�âºß{â�àÁâäæ�ìºâ��nçeê7à�á{ç`ãÁæVó�çé�æPðIæ>ãÏâ�ß �^âºî{÷ � ÷

àÁá{ç&ìºâºß{ç[ã�í�ô � � � � ��� � � �� ÷ 7ßÃí�àÁáÃçnè�ìºâºß{çgíVô?êyç[ëVç[ì�í>å{ó�çnß,à/é`âºìºìÏïtç&à�á{çEâºßÃñ�í>èÁå]íVè9æÊà�âºíVß í�ô$óIæVã�ãñ�íVßtã²ç[èÁëÊæÊà�âºíVß6ñníVè�èÁçeñ�àÁâºí>ßÃã�â�ß�àÁáÃçjå{è�ç[ãÁçnß,àIìºçnëVç[ì7ã²çnàgãÁñ9áÃçnó�ç�ô¾íVìºìºíÊé`âºß{îjâäêyç[æ>ã=í�ô �ÏíVò{è�ìºâ�í>ò�� � � �!�é`á{âäñ9ágé�çnè�ç$âºó�å{ìºç[ó�çnß,àÁçeê�ã²í�ô�æ�è`ï¹ðgæVßÃêEüyñ9á{ß{ç[âäêyçnè`æ�ßtê �(ìºçnâºß � ���� ÷

� Ü��gÖEÙ � � �7Ú�� ���7Ö=×�Ïá{çgæ�ò{àÁá{í>è�ã/îVè9æÊàÁçnô¾ò{ìºìºð�æVñ9ý¹ß{íÊé`ìºç[ê{îVçFàÁá{ç�ã²ò{åÃåtí>è²à�í�ôÏàÁá{ç �ç[åÃæ�èÁàÁó�ç[ß,à/í�ô �-ß{çnè�îVð $è�æVß>àFû»í{÷ � � � � õ � � � � �������� æVßÃê � � � � õ � � � ������� �� û?ü�� $è�æVß>à7û�í{÷ #��ü �� ������ ��� æ�ßÃê&û#�ü� $è�æVß>à-û»í{÷Rû# � õ �� ��� ÷!�Ãò{èÁàÁá{ç[èÁó�í>èÁç � é�ç7àÁátæ�ß{ý��Ïá{í>óIæVã üyñ9áÃß{çnâäêyç[èRô¾íVè�å{è�íÊë,âäêyâºß{î�àÁáÃç�ó�ò{ì�àÁâ�õîVè�âºêgñ�íyêyç�÷

� � �<�`Ø��7ÖEÜ ���� ��� Þ�÷ 7ìºçnâºß{íÊëjæVßÃê �7÷ F÷)ù-òÃñ9ýVçnà²à[÷�ÿ-íVó�å{òyà�âºß{î&üyò{è²ô�æ>ñ�ç ��çnßÃãÁâºíVßpé`â�à�á �»âºîVáyõ²ø�è9êyçnè �(çnè�ß{çnìäãn÷Þ¤ß � ÷ �$æÊô¾ç��=æ�ßtê �=÷�ø$ã²áÃâ�óIæ � çeêyâ�àÁíVè9ã ���������� ����������������������� �!�"#�$�!%������&�!%&')(+* ,.-/� ���01,� �324� ,5-+0/�6%&����� �!%&'874'901���;:<* �!%&,=���#� � ëVíVìºò{ó�ç$Þ � åÃæ�î>ç[ã � 3����� �:æ�ýVç �:æ�áÃí,ç � ÿ � üyçnåyà�çnó�ïtç[è�� � �� ÷

� � � (÷¹üT÷ 7ìºó�î>èÁç[ß � > ÷���÷���ç[ìºì � ù^÷yÿ-íVìºçnìºìäæ � ��÷���÷���íÊé-ç[ì�ì � æVßÃê � ÷ � ÷�ú�ç[ìäñ�íVó�çV÷�qÿ-íVßtã²ç[èÁëÊæÊà�âºëVç»ê{æVåyàÁâºëVçpù è�í � çeñ�àÁâºí>ß �jç�àÁáÃí¹ê�ô¾íVèBàÁá{ç ��æVèÁâäæ�ïÃì�ç $çnßtã²â�à¤ð�Þ¤ßÃñ�í>ó�å{èÁçeãÁãÁâºï{ìºç&û�æPë¹âºçnèÁõ¦ü,à�íVý>ç[ã�-þ,òÃæÊà�âºíVßÃãn÷@? �&01�$�!% 'A�B��24� ,5-+0/�6%&����� �C% 'D�@��* ����E� � � � � " � 3��������� � � ÷

� � � (÷)ü]÷$7ìºó�îVè�çnß �+> ÷ ��÷ �Ïçnìºì � ù^÷�ÿ-íVìºçnìºìäæ � æVßÃê �$÷$�jæ�èÁàÁáÃæVìºçnè[÷ !ÿ�æ�èÁàÁçeã²âäæ�ß $è�âäêBù-èÁí � ç[ñ�àÁâºíVß�Eç�à�á{íyê �tíVè��Ïá{ç�Þ¤ßÃñníVó�å{è�ç[ã�ãÁâ�ïÃì�ç � ò{ìºçnè �-þ,òÃæ�àÁâºíVßtã�âºß�ÿ-íVó�å{ìºç�� $ç[íVó�ç�à�èÁâºçeã�÷ (� �F.G ?1H (!�#� H2I�&,.-!0/� H � ��� � � � " � � � !3����� ����� �� ÷

� ��� (÷TüT÷ 7ìºó�î>èÁç[ß �/> ÷ �$÷ �Ïçnìºì � æ�ßÃêBú ÷� F÷Tü��[ð¹óIñ#�[æVý]÷ û»òÃó�çnè�âºñ[æ�ì��EçnàÁá{íyê�ô¾í>è`àÁá{ç�Þ¤ßÃñ�í>ó=õå{èÁçeãÁãÁâºï{ìºç7û�æPë¹âºçnèÁõ¤ü,àÁí>ýVç[ã �-þ,òÃæ�àÁâºíVßÃã �`æVãÁç[êIí>ß�æVß »å{å{è�í��¹âºóIæ�àÁç7ù-èÁí � ç[ñ�à�âºíVß÷ (1 BF5G ?1H (!�#� H2I�&,.-!0/� H � ��� �!� � " ���� 3�� � � ��� � �� ÷

� �

Page 23: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

� � � > ÷ �$÷��Ïçnìºì � ù^÷-ÿ-í>ì�ç[ìºìºæ � æVßÃê ��÷�� ÷ $ìäæ�V÷ Ñü¹çeñ�í>ßÃê¹õ²ø�è�êyç[èFù-èÁí � ç[ñ�àÁâºíVß �EçnàÁá{íyê6ô¾íVèFàÁá{çÞ¤ßÃñ�í>ó�å{èÁçeãÁãÁâºï{ìºçFû$æPë¹â�ç[è²õ¤ü,à�íVýVçeã �-þ,òÃæ�àÁâºíVßÃãn÷ ? � 01�$�C% ' ��� 2I�&,.-!0/�"%������&�!%&' �@��* ����E� � ��� " ������3� ����$�� �� ÷

� � � > ÷ �$÷ �Ïçnìºì � ù^÷:ÿ-í>ìºçnìºìäæ � æVßÃê � ÷ ��÷ ��íÊé-ç[ì�ìh÷ »ßpç���ñnâºçnß,à?ãÁç[ñníVßÃêyõhí>è�êyç[è»å{è�í � çeñ�à�â�í>ßjó�ç�à�á{íyêô¾íVè`ë¹âäãÁñníVòÃã�âºßÃñ�í>ó�å{è�ç[ã�ã²âºï{ìºçÏöÃíÊé?÷^Þ¤ß ���B� ���&������� �B� ���� ��� ��� F BF5F 24� ,5-+0/�6%&����� �C% 'D7 ' 01���: * �C% ,����E� 24� ���$E��E�C�� � åÃæVîVçeã ����� 3������ � �»í>ß{íVìºò{ìºò � �7Þ �/> ò{ß{ç �� õ ��� �� � � ÷

� ��� > ÷ �$÷ �Ïç[ì�ìæVßÃê /÷ � ÷��jæVè�ñnòÃã�÷ üyç[ñ�í>ßÃê¹õ²ø�è�ê{çnèÏù è�í � çeñ�àÁâºí>ß �Eç�à�á{íyêBô¾íVè ��æVèÁâäæ�ïÃì�ç� �ç[ßÃãÁâ�à¤ð�:ìºíÊé7ãn÷@? �&01�$�!% ')����24� ,5-+0/�6%&����� �C% '�����* ����#� � ���� " ���� 3���� ��� �� � � ÷

� � � (÷ ��í>ò{è�ì�âºí>ò��T÷ �ÿ-íVò{åÃì�çeê �)çnë>çnì�õ¤ü¹ç�à �RíVìºò{ó�çnõ¤ø»ôäõ �^ìºò{âäê 7ìºîVí>èÁâ�à�á{ó!ô¾íVè#�)è9æVñ9ý¹âºß{î �jæÊà�çnè�âºæVìÞ¤ß>à�çnèÁô�æVñnç[ãn÷�� �B� F.� � 0 %&'=24� ���$E��E�C�� �B� ���� 2A7): (!� �E�����* �B� 24% �C%���% � �ÏæVß�� � 7ìºï]çnèÁà�æ �ÿ�æ�ßÃæ>ê{æ �/> ò{ß{ç ��� õ ��� � �� �� ÷

� � > ÷ /÷ �Ïè�æ>ñ9ý,ïÃâ�ìºì � /÷ �$÷ �(íVàÁá{ç � æVßÃêAÿ»÷*6TçnóIæVñ9á�÷ ÿ-íVß,àÁâºß¹ò{ò{ó �Eç�à�á{íyê ô¾í>è �Eíyêyçnìºâºß{îü¹ò{èÁô�æVñ�ç ��çnßÃãÁâºíVß÷ ? � 01�$�C% 'A�B��24� ,5-+0/�6%&����� �!%&'D��� * ����E� � ����� " �����!3��������� � � ÷

� ��� � ù^÷ F÷� $è9æ�nâºß~æ�ßtê û�÷ ��÷ �»çnâäêT÷�� *��&�B���&* �!%&,=��� (!�"%��$��' � ��* ÷�ÿ�æ�ó�ï{è�âäêyîVç »ß{âºë>çnè9ã²â�à¤ðEù è�ç[ã�ã �ÿ�æ�ó�ï{èÁâäêyî>ç � �� ��� ÷

� � � � /÷ �7÷��^ð,ô¾ç � �7÷,ü]÷yø�è�æVß � æ�ßtê �~÷ > ÷��Ãè�â�àÁà�ãn÷Tü¹òÃè²ô�æ>ñ�çÏàÁç[ßÃãÁâ�í>ß�æVßÃê �»âäã�ñ�í>ãÁâ�à¤ð?é`â�à�á �:æ�î>è�æVß{îVâäæ�ß�»ðyêyè�í¹ê{ð,ßtæ�ó�âäñnã íVßBæ ��èÁâäæ�ßÃîVò{ìäæ�è��jç[ãÁá÷@? �&01�$�!% 'A�B��2I�&,.-!0/�"%������&�!%&'D��� * �����#� � ��� " ��� 3��������� ���� ÷

� � ��� > ÷ $ìºâºó�ó � ø/÷ �jñ��ÏèÁð,æ�ß � �(÷ �Eç[ß{âºýVí�� � æ�ßÃê F÷�/÷$ü¹áÃæVèÁå�÷ �tèÁí>ß,à�à�è�æ>ñ9ý¹â�ßÃî�æ�åÃå{ìºâ�çeê à�íè�æPð¹ìºçnâºîVá{õNà9æPð,ìºí>è:âºßÃã²à�æVï{âºìºâ�à¤ðV÷ (� �F.G ?1H (!�#� H (+�6%&� H 2I�&,.-!0/� H � � � � � " �� �!3��� � � �� ���� ÷

� �� � /÷� $ò{çnð���çnè ��> ÷ �)â � �÷,û$æVêyâºó � ��÷¹üyñ[æ�è9êyíÊëVç[ì�ìºâ � æ�ßÃêIüT÷ 6TæVìºç[ãÁý,âh÷ �Rí>ìºò{ó�ç�õ´í�ôäõ �^ì�òÃâºê�Þ¤ß,àÁçnèÁô�æVñnç�)è9æVñ9ý¹âºß{î?é`â�àÁágü¹ó�í¹í�à�á{ç[ê�ü¹òÃè²ô�æ>ñ�ç�ü,à�èÁçeãÁã �Eç�à�á{íyê{ãRô¾í>è �Ïá{è�çnçnõ´ê{â�ó�ç[ßÃãÁâ�í>ßÃæ�ì��^ìºíÊé7ã�÷I? � 01�$�!%&'����24� ,5-+0/�6%&����� �C% '�����* ����#� � �� � " � �� 3������� �� � ÷

� ��� � �-÷!��÷!��æ�è�ìºíÊé æ�ßtê > ÷ �»÷�ú çnìäñ9á÷Bû�ò{ó�çnè�âäñnæVì^ñ[æ�ìäñ�òÃìºæ�àÁâºíVßEíVôÏàÁâºó�ç�õ¦êyçnå]çnóIê{çnß,à$ë¹âäã�ñ�íVòtã?â�ß{õñ�íVó�åÃèÁçeãÁãÁâºï{ìºç/öÃíÊé íVô`öÃò{âäê~é`â�àÁá æjô¾èÁç[çgâºß,àÁç[è²ô�æ>ñ�ç�÷�� ������ * �����#� �B��7 ' 01���&� � � " � ��� ��3� ��� ���� ���� ÷

� ����� ÿ-á{íVè�âºß (÷ > ÷ ß¹ò{ó�ç[èÁâäñ[æ�ì�ó�çnàÁá{íyê ô¾í>è�ãÁíVìºë¹âºß{î&â�ßtñ�íVó�åÃèÁçeãÁãÁâºï{ìºç/ë¹âäãÁñníVòÃã�öÃíÊé!å{èÁí>ï{ìºçnóIãn÷? �&01�$�!%&'8���;24� ,5-+0/�6%&����� �C% 'A����* ����#� � � " ���!3������ �� ���� ÷

� �� � �-÷ �(âäñ9ý¹âºß{îVç[èn÷ 7ìºîVç[ï{è�æVâäñ-ó�ò{ì�àÁâºî>èÁâäê(ô¾í>è-êyâäã�ñ�è�ç�à�ç7çnìºìºâºåyàÁâäñÏã²çeñ�í>ßÃê=íVè9êyçnè-å{è�íVï{ìºçnóIãn÷$�)çeñ9á{ß{âäñnæVì�7çnå]íVèÁà � ���� Þ¤ßÃã²àÁâ�àÁòyà`ô� òÃè �jæÊà�á{çnóIæ�àÁâºý � �ß{âºëVç[è�ãÁâ�à� æÊà �)âºß � � �� � �� ÷� ���� ��÷ ��æ�ô�æ�ò{è�âºç � ÿ»÷,û$æ�è9êyíVß{ç � ��÷{ü{ñnæ�è9êyíÊë>çnìºìºâ � ü]÷ 6TæVì�çeã²ý¹â � æVßÃê F÷ 6Tæ�ß{çnà²à�âN÷ �jí¹ê{çnìºìºâ�ßÃî"�jçnè�îVâºß{î

�Ãè9æ�îVó�ç[ß,à�æÊà�âºíVß/âºß��Eò{ì�à�â�åÃáÃæVãÁç �:ìºíÊé7ã^é`â�àÁáIü (� � �,�(÷4? � 01�$�C% 'C���<24�&,.-!0/�"%������&�!% '+��� * ����E� ������ " ��� 3�� � � ��� � �� ÷

� ��� � �(÷ > ÷ ��ç �Rç[þ,ò{çFæ�ßÃê 6�÷ �)âh÷ �Ïá{ç(Þ¤ó�ó�çnè9ãÁç[êgÞ¤ß>à�çnèÁô�æVñnç �Eç�à�á{íyê&ô¾íVè �-ìºì�âºåyà�âäñ �-þ,òÃæ�àÁâºíVßtãÏé`â�à�á �âäã�ñ�íVß,à�â�ß¹ò{í>òÃã:ÿ-í¹ç���ñnâºçnß,à�ã:æ�ßÃê�ü¹âºß{îVòÃìºæVè�ü¹í>ò{è�ñnç[ãn÷ (1 BF5G ? � 01�$�C% ' �&��� 01,�E�$���%&'�F.�!% '9* ����� ������ � � " ������ 3������� ��� � ÷

� �� � �(÷ > ÷ �)ç �Rçeþ,ò{çgæ�ßtê 6�÷ ��âh÷EÞ¤ó�ó�çnè9ã²çeê�Þ¤ß,à�çnèÁô�æVñ�ç �jç�àÁáÃí¹êÃã�ô¾í>è�ü,à�íVýVçeã �^ìºíÊé�é`â�à�á � ìäæVã²àÁâäñ��í>ò{ßÃê{æVèÁâºçeã?íVèFü¹òÃè²ô�æ>ñ�ç ��çnßÃãÁâºíVß÷ (� �F.G ? � 01�$�C% ' � � (C�E��#�+��� � � 24� ,5-+0/��������� ��� " � �� 3 ����� ��� � �� ÷

� ��� � (÷ �jæPð>í{÷ �ÏáÃç �]æVã²àgü¹íVìºòyà�â�í>ß í�ô$ùRíVâäã�ã²í>ß$ ã/æ�ßÃê6àÁá{ç �ÏâºáÃæVèÁó�í>ß{âäñ ��þ,òÃæÊà�â�í>ß íVß{Þ¤èÁè�çnî>ò{ìäæ�è�7çnî>â�í>ßÃãn÷ (1 BF.G ? � 01�$�C% 'A�&��� 01,�#�$����%&'CF.�!%&' * ���� � � ���!� � " �����!3�� ����� ���� ÷� ��� � ÿ»÷]ü]÷{ùRç[ãÁý¹âºß÷:û�ò{ó�çnè�âäñnæVì]æVßÃæ�ìºðyã²âäã í�ô�ïÃì�í¹íyêgöÃíÊéqâºßIà�á{ç(á{çeæ�èÁàn÷ ? � 01�$�!%&'8���;24�&,.-!0/�"%������&�!% '

����* ����#� ����� " � � � 3���� � ��� ���� ÷� � ��� �(÷ùRçnð¹èÁçnà�æ�ßtê �$÷ F÷���æPð¹ìºíVè[÷ 24� ,5-+0/�6%&����� �C% '4,�#����� � ���$�&����01����� ��� ÷�ü¹å{è�âºß{îVç[è»ãÁçnè�â�çeã7âºß

ñ�íVó�åÃòyà�æ�àÁâºíVßÃæVìtåÃá,ðyãÁâäñnãn÷{üyå{èÁâºß{î>çnèÁõ �Rç[èÁìäæVî � û�çné���íVè�ý � ��çnâäêyç[ì�ï]çnè�î � �Ïçnè�ìºâ�ß � �� ���� ÷

���

Page 24: A Projection Method for Two-Phase Incompressible Flow with ...webdoc.sub.gwdg.de/ebook/e/2003/zib/reports/ZR-00-17.pdf · Takustraße 7 D-14195 Berlin-Dahlem Germany Konrad-Zuse-Zentrum

� ��� � (÷yù è�í>ãÁåtç[èÁçnà²à�âN÷��EíVàÁâºíVß�íVôà¤é-í�ã²òÃåtç[èÁâºó�å]í>ãÁç[ê�ë¹âºã�ñ�í>òÃã^öÃòÃâºêÃã�÷ �@��* ����E� ��� 7 ' 01��� � ���� " ��� ���3��������� �� ��� ÷

� �� � �7÷� F÷¹ù-òÃñ9ýVçnà � (÷Ãü]÷�7ìºó�îVè�çnß ��> ÷��$÷��Ïç[ì�ì � /÷ ��÷�jæVè�ñnòÃã � æ�ßÃê�ú ÷ > ÷ �7âäêyçnè[÷ ��â�î>áyõ²ø�è�êyç[èù è�í � ç[ñ�àÁâºíVß �Eç�à�á{íyê ô¾íVè ��è�æ>ñ9ý,âºß{î �^ì�òÃâºê6Þ¤ß,àÁç[è²ô�æ>ñ�ç[ã=âºß ��æ�è�âäæ�ï{ìºç $çnßtã²â�à¤ð~Þ¤ßÃñníVó�å{è�ç[ã�ãÁâ�ïÃì�ç�:ìºíÊé7ãn÷@? �&01�$�!% ')����24� ,5-+0/�6%&����� �C% '�����* ����#� � ��� � " ��� 3���� � � �� � �� ÷

� ����� ú ÷ �(÷��7âäêyçnè � F÷��$÷ �(íVàÁá{ç � ü]÷ > ÷ �Eí>ã�ã²í ��> ÷���÷{ÿ-çnè�òyà²à�â � æ�ßtê > ÷VÞ�÷��»íyñ9áÃã²àÁç[â�ß�÷ »ñ[ñ�ò{è9æÊà�ç»ü¹í�õì�ò{àÁâºíVß »ìºîVíVè�â�àÁá{óIãRô¾íVèÏÞ¤ßÃñníVó�å{è�ç[ã�ã²âºï{ìºç �Eò{ì�à�â�åÃáÃæVãÁç �^ìºíÊé7ã�÷ ���B� ���&������� �B� ���� � � �B�;F BF5FF #�B� �"-/%��=(!�#��E�C� G #������� % �C� ��� � � �$� �!� 7Þ `õ �� õ ���� �� �7çnß{í � û � � > æVß÷ �� �� ÷

� ��� � F÷-�7ðyã²ý¹âºß æ�ßÃê ��÷! F÷ �)çeæ�ìh÷�û�ò{ó�çnè�âäñnæVì�ü¹í>ì�ò{àÁâºíVß~íVô �Ãè�çnçnõ �ÏíVòÃßÃê{æ�è�ð�å{èÁí>ï{ìºçnóIã(âºß �^ìºò{âäê�Eç[ñ9áÃæVß{âäñnã � ù�æ�èÁà � ÷��Ïò{íÊð>æVßÃñ�ð,õ �è�âºëVç[ß �Eí�à�â�í>ß6íVô?æ ?æVã �Ïò{ï{ïÃì�ç �Ïá{è�íVòÃîVá�æ���ò{âºçeãÁñnçnß,à�)âäþ,ò{âäêT÷@? �&01�$�!% ')���.7 ' 01����G �#��%&� ���#� � � � � " �� !3 ����� �� �� ÷

� ����� ùRíVå{âºß{çnà=ü]÷-æVßÃê 6æ�ìºç[ãÁý¹âÏü]÷ 8ô¾èÁí>ß,à²õhàÁè9æVñ9ý¹âºß{î�æ�ìºî>íVè�â�à�á{ó+ô¾íVè�æVñ[ñ�ò{è9æÊà�çBè�çnå{è�ç[ãÁçnß,à9æÊàÁâºí>ß~í�ôã²ò{èÁô�æVñnçÏàÁç[ßÃã²âºí>ß÷ $�+�6E�$�C%&����� �C% ' ? �&01�$�!%&'��$�&��� 01,�#�$����%&'/G ��������&�.��� 7 ' 01��� � � ��� � � � " ����!3 � ������ � ÷

� ��� � �(÷)üyñnæVè�êyíÊë>çnìºìºâæ�ßÃê�üT÷ 6æ�ìºç[ãÁý¹âN÷ $âºèÁçeñ�à»û�ò{ó�çnè�âäñnæ�ì:ü¹âºóFò{ìäæ�àÁâºíVßBí�ô��Ãè�çnçnõ¦üyò{è²ô�æ>ñ�ç/æVßÃêEÞ¤ß,à�çnèÁõô�æVñ�âäæVì �^ìºíÊé?÷ F.� �+0 H�� �� H 7 ' 01���;G �#� H � �� " �����3�� ������ � ÷

� �� � �$÷¹üyñ9á{ßÃçnâäêyçnè[÷ 6]ò{è �^ìäæ�ó�ó�ç[ß¹ëVçnèÁô¾íVìºî>ò{ß{î`âºß�ãÁñ9á¹éÏæVñ9á{ý>íVó�å{è�ç[ã�ã²âºï{ìºç�ß(è�ç[æVý,àÁâºëVç[ß=ü,à�è� í>óFòÃß{îVç[ß÷: ����EE�E�6%&����� � � � � F.%��#��#� � ãÁò{ï{ó�â�à²à�ç[êT÷

� � � � �$÷�üyñ9á{ß{ç[âºê{çnè � û�÷ �Ïí�àÁà�æ � �=÷ > ÷� $çnè9æÊà�� � æ�ßtê �(÷ �(ìºç[â�ß�÷ ���¹àÁç[ßÃã²âºí>ßjí�ô �Ãß{â�àÁç�ëVí>ì�òÃó�ç/ñ�í>ó=õå{èÁçeãÁãÁâºï{ìºç�öÃíÊé"ã²í>ìºëVçnè9ã»àÁíEó�ò{ì�àÁâ�õ´ê{â�ó�ç[ßÃãÁâ�í>ßÃæÊì � ëÊæ�è�âäæ�ï{ìºç�êyçnßÃãÁâ�à¤ð �nç[èÁí �jæ>ñ9ápß¹ò{ó�ïtç[è?öÃíÊé7ãn÷? �&01�$�!%&'8���;24� ,5-+0/�6%&����� �C% 'A����* ����#� � ���� " ��� �!3��������� � ÷

� �� � �$÷Êüyñ9áÃß{çnâäêyç[è)æ�ßÃê��(÷ �(ìºç[â�ß�÷,ø�ëVçnè9ñ�í>ó�âºß{î óIæ>ãÁãìºí,ãÁãÁç[ãâºß ��çnëVç[ì�õ¦ü¹çnà²õ´ïÃæVãÁç[ê�âºß,àÁçnèÁô�æVñnç^àÁè9æVñ9ý¹âºß{îãÁñ9á{ç[ó�ç[ãn÷ (!�� �C�5 �!�"#�$�!%������&�!%&'/( * ,5-1�&���01, �&��7 ��� � �6�� � ' 01,�#�C�$�&� 24� ,5-+' � F�- -+' ���%&����� �+� ���B���$' E,�� % �C��� #�$�"-1������� E� �!> ò{ìºð �� õ �������� � � � ��ç[èÁìºâºß � $çnè�óIæ�ß¹ð�÷

� � ��� ��÷»ü¹ó�âºì � æ�ß{íÊëyãÁý,â � ��÷ �Eí>ãÁçnè � æ�ßtê �(÷��(ìºçnâºß÷ 5ñnæ�å{àÁò{è�âºß{î�õhàÁè9æVñ9ý¹âºß{î�á,ð¹ï{è�âäêAãÁñ9á{ç[ó�çEô¾íVèêyç�ötæVîVè9æÊà�â�í>ßgêyâäã�ñ�íVß,à�â�ß¹ò{â�à�â�çeã�÷ 24�&, �E01�$�����&� � ���&�$*�% �C��G ��� E'�'9�������$� � � � " �� � �� � �� �� ÷

� ��� � � ÷Vü¹òÃã�ã²óIæVß � (÷Vü]÷ 7ìºó�îVè�çnß � > ÷���÷�Ïçnìºì � ù^÷>ÿ-í>ì�ç[ìºìºæ � ��÷��/÷��íÊé-ç[ìºì � æ�ßÃê � ÷ ��÷Pú çnìäñ�í>ó�ç�÷ »ßæVê{æVåyàÁâºëVçRìºç[ëVçnì�ãÁç�à^æ�å{åÃèÁí,æVñ9á�ô¾íVè)âºßÃñníVó�å{è�ç[ã�ãÁâ�ïÃì�ç�à¤é-íVõhå{átæVãÁç^öÃíÊé?÷/? � 01�$�!%&'��B�.24�&,.-!0/�"%������&�!% '����* ����#� � � � � " ����3�������� � ÷

� ��� � � ÷ ü¹òÃã�ã²óIæVß æ�ßÃê �»÷��]æÊàÁç[ó�âh÷ 7ß�ç#�Iñ�âºçnß,à � âºß,àÁç[è²ô�æ>ñ�çIåÃèÁçeã²ç[èÁë¹âºß{î&ìºçnëVç[ì-ãÁç�à�èÁçnõ´êyâäã²à�æVßÃñ�âºß{îæ�ìºîVí>èÁâ�àÁáÃó"æVßÃêgâ�à9ã`æ�å{å{ìºâäñnæ�àÁâºíVß�à�í�âºß,àÁç[è²ô�æ>ñ�âäæ�ì]âºßÃñ�í>ó�å{è�ç[ã�ã²âºï{ìºç�öÃò{âäêgöÃíÊé?÷ (1 BF5G ? �&01�$�!% 'A�B�(!�#��E�!��� ��� 24� ,5-+0/��������� ��� � � � " �������3 ���� ���� �� � ÷

� ����� � ÷Ãü¹òÃã�ã²óIæVß � �»÷ �tæÊà�çnó�â � ù^÷yü¹ó�ç[èÁç[ýÊæ � æ�ßÃê�ü]÷tø$ã²á{ç[èn÷ 7ßBÞ¤ó�å{èÁíÊë>ç[ê �)ç[ëVç[ìüyç�à �Eç�à�á{íyê�ô¾íVèÞ¤ßÃñ�í>ó�å{èÁçeãÁãÁâºï{ìºç �Ïé�í�õ´ù áÃæ>ã²ç��:ìºíÊé7ãn÷ 24�&,.-!0/�"#�$��� 7 ' 01���&� � ��� " �����!3 ��� ��� �� � � ÷

� ��� � � ÷Ãü¹òÃã�ã²óIæVß � ü¹ó�ç[èÁç[ýÊæ(ù^÷ � æ�ßtêBø$ã²áÃçnè�üT÷ qìºç[ëVçnìtã²çnà�æ�åÃå{èÁí,æVñ9á=ô¾í>è-ñ�í>ó�å{òyà�â�ßÃî�ã²í>ì�ò{àÁâºíVßÃã�í�ôâ�ßtñ�íVó�åÃèÁçeãÁãÁâºï{ìºçÏà¤é-íVõhå{átæVãÁç»öÃíÊé?÷ ? � 01�$�C% '8�B�;24� ,5-+0/�6%&����� �!%&'D��� * ����E� � ��� � " ����� 3���� ���� � �� ÷

� ����� ù^÷ ��çnè�á{í¹çnëVç[ß÷�� ��� � 01,�E�$���E�#��#��� E���$%&���B#���E01���5E����#���+01�������&�37 ' %&,=,�E�����B� �!�"#����#�� ' #���!#� G %��#�����)% ��' ÷ $âºã�ãÁçnèÁà�æÊà�âºíVß � ��ú � �#»æVñ9áÃçnß ���� � � ÷

� � � � F÷ ��èÁð¹î>îVëÊæVãÁíVß � �$÷!�Ïò{ßÃß{çnè � ø/÷ �-ï{è�æ�à � æ�ßÃê~ú ÷ �:æ�ò{ï]çnè[÷ ÿ-í>ó�å{òyà�æ�àÁâºíVßtã(í�ô �EòÃì�à�âºå{áÃæVãÁç�:ìºíÊé7ãÏï¹ðgæ �^âºß{â�àÁç $â �Tçnè�çnßtñ�ç � �Ãè�íVß,à �)è9æVñ9ý¹âºß{î �jç�àÁáÃí¹ê÷ÃÞ�÷��EòÃì�à�â�õ �^ìºò{âäê �^ìºíÊé7ã�÷ � àÁá&ñ�í>ó=õå{òyà�æ�àÁâºíVßtæ�ì�öÃò{âäê êyð¹ßÃæ�ó�âäñnãFìºç[ñ�àÁò{è�ç�ãÁçnè�âºç[ã � �RíVß ��æVèÁóIæVß~Þ¤ßÃã²àÁâ�àÁòyà�ç�ô¾íVè �^ìºò{âäê $ð,ßtæ�ó�âäñnã ��� � � õ ��� ÷

� � � ü]÷�ø/÷� »ß¹ë>çnè9êyâ>æVßÃê F÷ �)è�ð¹îVîVëÊæ>ã²í>ß÷� �tèÁí>ß>à �)è9æVñ9ý¹âºß{î �EçnàÁá{íyê?ô¾í>è ��âºã�ñ�í>òÃã � Þ¤ßÃñníVó�å{è�ç[ã�ãÁâ�ïÃì�ç�Eò{ì�àÁâ�õ �:ìºò{âäê �^ìºíÊé?÷ ? �&01�$�!%&'8���;24� ,5-+0/�6%&����� �C% 'A����* ����#� � ��� � " ���!3���� � �� � ÷

��