PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact...
Transcript of PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact...
![Page 1: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/1.jpg)
![Page 2: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/2.jpg)
![Page 3: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/3.jpg)
![Page 4: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/4.jpg)
(Arch. Rat. Mech. Ana. 76, 97-133 (1981))
![Page 5: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/5.jpg)
![Page 6: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/6.jpg)
![Page 7: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/7.jpg)
![Page 8: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/8.jpg)
![Page 9: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/9.jpg)
![Page 10: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/10.jpg)
![Page 11: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/11.jpg)
Georg Friedrich Bernhard Riemann (1826 – 1866)
Peter David Lax (1926--)
Riemann (1860)
Lax (1957)
Mathematical Theory of Conservation Laws
Commun. Pure Appl. Math., 10 (1957)
![Page 12: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/12.jpg)
Simple Waves In the case the characteristic field is genuinely nonlinear
Shock wave :
Rarefaction Wave :
Rankine-Hugoniot Condition + Entropy Condition
![Page 13: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/13.jpg)
Simple Waves : In the case the characteristic field is linearly degenerate
Contact discontinuity :
General Riemann Solution
A linear superposition of simple waves
![Page 14: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/14.jpg)
An example
![Page 15: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/15.jpg)
Asymptotic state under viscous effect
・Rarefaction wave
・Shock wave Viscous shock wave
・Contact discontinuity Viscous contact wave
Viscous shock wave :
Viscous contact wave:
Traveling wave relaxed by viscosity
Diffusion wave relaxed by viscosity
Rarefaction wave
![Page 16: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/16.jpg)
An example
Riemann Solution:
Asymptotic Solution:
![Page 17: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/17.jpg)
Known results on Viscous and Heat-conductive case :
Rarefaction + (Rarefaction) Kawashima-Nishihara-M (1986)
Single Shock Kawashima-M (1985) (zero mass initial perturbations)
Shock +Shock
Rarefaction+Contact discontinuity+Shock Open
Liu (1997) Zumbrum (2004) Liu-Zen (2009)
Huang-M (2009)
Contact discontinuity +Shocks Open
Contact discontinuity+Rarefactions Huang-Li-M (2010)
Single Contact discontinuity
Huang-Xin-Yang (2008)
Huanag-Xin-M (2006) (zero mass initial perturbation)
Rarefaction+Shock Open
![Page 18: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/18.jpg)
![Page 19: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/19.jpg)
![Page 20: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/20.jpg)
![Page 21: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/21.jpg)
![Page 22: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/22.jpg)
![Page 23: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/23.jpg)
![Page 24: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/24.jpg)
![Page 25: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/25.jpg)
![Page 26: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/26.jpg)
![Page 27: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/27.jpg)
![Page 28: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/28.jpg)
![Page 29: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/29.jpg)
![Page 30: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/30.jpg)
![Page 31: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/31.jpg)
![Page 32: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/32.jpg)
![Page 33: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/33.jpg)
![Page 34: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/34.jpg)
![Page 35: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/35.jpg)
![Page 36: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/36.jpg)
![Page 37: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/37.jpg)
![Page 38: PowerPoint プレゼンテーション › ~akitaka › pdesemi.pdfContact discontinuity+Rarefactions Huang-Li-M (2010) Single Contact discontinuity Huang-Xin-Yang (2008) Huanag-Xin-M](https://reader033.fdocuments.net/reader033/viewer/2022053000/5f04681c7e708231d40dd0eb/html5/thumbnails/38.jpg)
Thank You!