Three Level Fourier Spectral Approximations for Fluid Flow with Low Mach Number

1. INTROlX_?CTION

iw ~+(C’-V)L’+QP-vO’U=f. x E $1, t E (0, T] ,

U( I-, 0) = U,,( s) , P( x,0) = P,)(X).

x E 12, (1.1)

whc~rc~ U is tllch r,-dim~nsionul sped vector, tl = 2 or 3, P is the ratio of the

pressure to the density, and v > 0 is the kinetic viscosity coefficient. Assume fi = (0,27r)“; the given function U,,(s), P(,( s), and f( K, t ) liavc~ period 27~

for all E,,. Due to its high or&r ilccuriicy, speotnd approxin~ation has now lwcorne ii

powc~rfd dtc~rnutivc~ to classicul finite diffcmncc~ or finite clement rnttthod in

numerical solutions of partiul cliffcmmtial ecpations [2]. (ho Benp and Ch

132 A. BASHID, C. WEIMING, AND G. BENYU

Weiming [3] first proposed a class of ful!y discrete spectral approximation to (I.D. It has been strictly proven that these approximations are of conver- gence rate O(r + M-“1, where (z + I)” is the number of unknowns, and s depends only on the smoothness of the exact solution of (1.1). Compared with the classical finite difference scheme (see [4]), where the convergence rate is O(I + h”), the spectral approximations are e+ected to producg more accu- rate numerical solutions. Thus, in [5] we improve the work of [3] to give a pseudospectral approsimation to solve (1.1). It is easier to use practical implementation; it still pertains to the conservation laws, the stability, and the convergence, attributing to the introduction of the filtering technique of [6].

the “infinite order” of convergence in spatial spectral or approximation* the accuracies in the temporal finite differ-

roxrmations seem rather unsatisfactory. Some authors combined the roximations with high-order time discretization techniques, e.g., r-corrector method [‘I’], the Runge-Kutta method [$I, or even

spectral method in time directions [9]. However, they were restric- me simple problems or merely computations; there is hardly any

rigorous analysis about them for nonlinear problems. The aim of this paper is to take (1.1) as a model problem to analyze strictly

tral approximation in combination with a second order time differ- hnique. The essence of this approximation is to adopt the

icolson method for the linear terms, and the second order rth method for the nonlinear terms in time discretization. The

eme is of convergence rate O(7’ + BP). ntents of this paper are #arranged as follows. In Section 2, we

uce some notation and basic lemmas. The three le%Jel spectral scheme presented in Section 3. In the remaining two sections, the stability and the convergence of the scheme, as well as the

convergence rate, are analyzed rigorously.

2. NCTATION AND LEMMAS

Let

I,@) = {Z(U)+ C(6J;Hp(n)),

Zhasperiod2?rforallXq1~q~n),

Spectral Approximation of Fluid Flow

IlW)II” = (z(t), W))T

IIW) IL = m=lz(x,t)l, XEfI

IIZ(t)lll,x = max “~:“) II

169Gn .“I

Ill w lllx = max Ilwll=. Ogt<T

133

+ ILW IL z

Illw) IIL = max Ilz(t)llL=~ Ogt<T

Let I = (11, d2, . . . ,

..* + ‘y/2_ 1,) be n-multiple integer, 111 = t&f + 1: +

For a positive integer N, we define

and we denote by Px the I? projection operator fn>m L2(Q) on to VN . To approximate the nonlinear terms in (1.0, we define a bilinear operator

d( l , l ) as follows:

1 d(Z,Y) = (Y*V)Z + pY)Z.

It is easy to verify by integration by parts that

(d(z,Y),w) + (d(W,Y),z) = o.

Let T be the mesh spacing of variable t,

s,= tt (I =kT,O\<k\< ,

P-1)

W) = I(Z(t + 7) - Z(t)). 7

154 A. RASHID, C. WEIMING, AND G. BENYU

It is easy to see that

2(VZ(t),VZ,(t)) = (IZ(t)Q, - Tpt(t)I;. (2.3)

MMA 2.3. Assume that

(iI E(t 1 and p(t) are nonnegatit;e. nondecreasing fitnctions dejned

on s,, 63 d, and d, am nonnegative constants, (iii) for all t E S,,

E(t) <p(t) + $(dlE(t’) + d,E’(t’)), I’=7

64 andforcetiain t, E S,, p(tJexp(Zd,t,) G (d&l,).

Then for all t < t ,,

Spectral Approximation of Fluid Flow

3. THE SPECTRAL SCHEME

We consider first the generalized solution of (1.1) as follows:

135

I au t -$t)J 1 + ((U(t) qJ(t),w) + pqt,,q

+ v(VU(t),Viq = (f(t),W),

(~(t)J~) + ((U(t) l V)P(t),W) = 0,

WV E (h(n))” Xll(W

U(x,O) = U&), P(x,O) = PO(X), x E a. (34

The Fourier spectral method for (3.1) is to find b(t), p(t)) E (I$ )” X V& for all t E S,, such that

(u,(t),W) + (;d(u(t),u(t)) - +d(u(t - +u(t - r)),E\ . / + (V$(t),lW) + v(Viqt),VW) = (f(t)jv),

WE (VJ,

(p,WW) + 3 24 p(t)* u(t)) - ;d( p(t - r), u(t - r)).w)

= 0, VWEV,

40) = q&, 471 (0)) ,

( dr? P(O) = PNPW p(7) = PN P, + 7; 0) , ( 3.2)

where N,‘&(O) and aP/dt(O) are determined from UO, P, and the equations of (1.1).

136 A. RASI-IID, C. WEIMING, AND G. BENYU

REMARK 3.1. Noting that exp( i t x) a-e the eigenfunctions of V” with periodic boundary conditions, scheme (3.2) is actually an explicit scheme. Compared to the explicit two level scheme (scheme [5] with 6 = 01, the above scheme requires merely 4(2N + 1)” more storages, but no more computations, while achieving one order higher convergence rate in time direction.

4. THE STABILITY

Suppose that the initial values ~(0) and p(O), U(T), and P(T) in (3.2) have , 1? 1= and j5 *, respectively, and that right-hand terms in the first

aud the second equation have errors f and jj, respectively. Then the errors 17(tI_ i;(t), in t&I and p(t) satis+

(ii,(t),F) + (;d(u(t).C(t)) + ;d(ii(t),u(t) + ii(t))

1 - zd(u(t - T),ii(f - 7)) .

1 -Zd(G(t - ~),zi(t - 7) + ii(t - T)),=) + (V;(t),E)

3 1 +ed( j!(t),u(t) + ii(t)) - ?d(p(t - ~),ii(t - 7))

1 -zd(j5(t - +u(t - d +ii(t - T),w) = (g(t),\v), I

(‘vt’,W) E (V&j" X *fN, (4.1)

Spectral Approximation of Fluid Flow 137

Let E and a be undetermined positive constants. By taking the inner product in the equation of (4.1) with 2ii(t) + aG,(t), we have from (2.11, (2.3), and (i) of Lemma 2.2 that

(Ilc(t)II”), + .(, - 1 - 2~ - -&rN”)~~ii&)(lP + 22&i(t)f

+;u + a)(lqt)l:), + i 5 &i(t)II” j=l

+ 1 + q (Ilf(t)ll' +Ilf(t + T)II”), ( 1 where

W)

F,(t) = &l(u(t),ii(t)),ei(t) + an?,(t)),

El(t) = ~a+(d(ii(t),a(f)) + ii(t)),&(t)).

F3(t) = -$( ( tl t - 7),1L(t - 7)),2ii(tj + a@(t)),

F,(t) = -$4 ( ii t- 7),u(t - 7) + iqt - 7)),2ii(t) + abi,(t)),

F&) = (V;(t) ,2ii( t) + a7iiJ t)).

Next, by taking the inner product of the second equation (4.2) with 2p’(t) + a+(t), similarly we have,

(II m l12)t + +J - j=S

41jwl12 + 1 + ( ) q II g(t) II”, (4 4) .

138 A. BASHID, C. WEIMING, AND G. BENYU

where

Fi(t) = +(d( jQ).z@)) + i(t)), FJt)),

F,(t) = -;(d( p(t - r),ti(t - T)),2@) + a+#)), I

F,(t) = -f(d( ji(t - r),u(t - T) + ii(t - +,zi;(t) + txjqt)).

The combi~atien af (4.3) with (4~4) leads to

(Ilii(t)lf +IIji(t)Qt + + - 1 - 2E - fd)

+,(t)lf +[I &(t>II’) + z&i(t)f

+1 + a)(liqt)f), + i I;;- j=l

where

&(t)II +IIii(t)ll+ G(t), (4J9

Now% we estimate 131. We have from Lemma 2.2, and the Sobolev embedding theorem that

+ z Ill u Ill 1.2 + c EV

Ill u Ill 1.&(t) II*>

IF21 G +&(t)ll” + ca2TN2 lll~lll~,m~~~(t)~~2 E

Spectral Approximation of Fhid Fkm 139

ca%N n’ 2 + E IIW) II43

IF,1 < Eulqt - 4; + +qt)II”

+ f Ill 14 Ill 1.r. + c) Ill u Ill 1.&q t) II23

( ca%N’

+ ---&--- Ill u Ill 1.r + c

1 Ill 11 Ill I.$qt - 7) II23

I2-y < #,(t) 11” + ETpq f - 7) 11’ + * Ill 24 Ill ;.$q t) II” l

+ ““:” Ilii(t)II” xlll’;(t - T)II’ + ca2TN2 IIu’(t - T)II’ l

Ca2TN 2

+- E

111 u 111 :.zcll iJ( t - 7) 112,

IF,1 < ~uli?(t)f + .yaT”N’lli?,(t)II” + ;(l + a)11 p(t)ll’

+ g(1 + 41 iwl12

IF,1 < EYl?q t) 1; + ETll jq t) 11’ Ca2TN 2

+ - Ill p Ill :.I + c E Ill p Ill I.$qt) II”

+ ( c Ill p Ill t9.J I.= + c Ill p lil1.Jl p(t) 112, 1

l&l \< -11 p,(t) (1’ + ca2TN2

E III p III f!. 211 jq t) II’

ca2TN n + 2

+ E IIW) Il7l pw IV?

I&I < wlqt - 4 It + -II jw) II”

140

+

+

A. RASI-IID, C. WEIMING, AND G. I3ENYU

z- III p Ill 1 ,Tx: + c 4% 1

Ill F Ill I.TI/ jq t) II”,

(

C&M"

e Ill p Ill I.r + c 1

Ill p Ill *.xllqt - 7)llf

3 + E$$(t - T)II’ + - c7N2 Ill II Ill f.fll jq t) 11’ E

CTp+ 3

+ E

jqt)(f! xllqt - T)II’

C&P -e II i G t-

c T) II’ll jq t - 7) 11’

+ E

11 ti 111 f_J ji( t - 7) Ii’.

By putting the previous estimations into (4.4), we get

- +qt - T)li” +IIjQt - T)f) + ?Jv(l - E)lqt)l; .

- Bw(Jll(t - T)Q + yi1 + 4(117,it)l;),

< A( 11 tq t) 11” + II j5( t) 11” i- IIii( t - 7) il” + II jq t - 7) II’)

+qllw - dl’) +Il~i~)ll’)(bq~)II’ +IIjwll”

+@(t - T)II” +lli(t - 7)112) + G(t), (4 6) .

Spectral Approximation of Fluid Flow 141

where

Let T = O( N-” ), s 2 2, and E be suitably small; then we choose the undetermined parameter a as follows,

VTN2 I r+1+6~+-.

aa 2 I I 1+ 2mN’C7

,

EV

which actwally means that

a -l-6e- 4(1 + a) - EV

Thus, we have from (4.6) that,

(Ilwll +I1 ii(t)l12), + b-1 + ~!$4wII’ +I1 ?zwIIP)

- ~~(lliqt - T)li2 +I1 jgt - T) 11’)

+ 2V(l - +@)I; + ;(I + a)(lW) I;),

G A(IIW II? + II p(t) IT + IIW - 7) II” + II pii - 7) II’)

+ B(llii(t - T)ll” +llii(t)ll”)(llii(t)ll’ +ll jS(t)/l’ +llii(t - r)ll”

+ II jw - 7) II”) + G(O* (4 7) .

142 A. BASHID, C. WEIMING, AND G. BENYU

E(t)) =Ilii(t)ll’+l~~(t)l~” + a(1 + a)ll’;(t)l;

+ rtg (‘,7(lliit(tr)I12 +I1 jqt8)(12) + vlqty;), t’=o

jgO)II’) + ea&i(O)I;

2 + II j?(O) II” -I- 11 fi( 7) II2 + 11 jq 7) II’)

-I +IIP(o)II’+IIwlr’+lljwII”)

+ ryqr’). t’=t

ming up (d.‘?‘) for t ’ E S,, T < E’ < I - T, we have

E(t) \( p(t) -I- rtc(2AE(t’) + BE’(F)). t”=t

By applying Lemma (2.31, we fkally reach the estimation regarding the generalized stability as follows,

E(t) 4 p(t)exp(Ut),

2 A p(t,)exg(4At) < 7 = CT-‘N-~-“.

THEOREM 4.1. Assume that in scheme (3.2) T = O( N-“1, s >, 2, and there exists a t , E S,, such that

Spectral Approximation cf Fluid Flow 143

with M, being suitably snd. Then we have for all t E S,, and r < t < t, that

E(t) G p(t)eq@At).

5. THE CONVERGENCE

In this section, we deal with the convergence of (3.2). Let

U”(t) = P&(t), P”(t) = Ph?P(t);

then we obtain from (3.1) that

(v,“(t),~) + ( ;d(Us(t), v”(t)) - ;d(V(t - r), v”(t - r),v)

+ (V$yt),\fq + v(Vti”(t),VW)

= (E, + E, + E, + VE, + E,,Iv)

+ Y(VS~, v\W), VW E (VJ,

(P,“(t),W) + (;d(P’(t),Ul(t)) - ;d(P’(t - r),uN(t - .),w)

= ( E6 + E; + E,,W)

V(O) = PJJ@ P”(0) = PNPO,

V(T) = P#T/T(T), PN(r) = PNP(T),

E,(t) =f(t + r/2) -j(t) dU”

ut) = W) - -j-(t + 7/q,

3 1 W) = zd(U(t),U(t)) - Zd(U(t - r),U(t - r))

- d(u(t + r/2), U(t + r/2)),

Ed(t) = P^(t) - P(t + r/2),

144 A. BASHID, C. WEIMING, AND G. BENYU

E,(t) = l?(t) - U(t + T/2),

dPS E,(t) = P#( t) - - at (t + T/2).

3 1 E;(t) = p(P(t)%U(t)) - p(P(t - ‘P)Jqt - 7))

_- d( P(t + i/Z)= U(t + T/2)).

3 3 qt),U(t) - R,U(t)) - p(U(t), &lb(t))

+ gd(R,U(t - 7)_ U( t - 7) - R,V(t - T))

3 3 &do = - ,ww(t). W) - ww) - ~n(p(t)* R,W))

I +pd(R,P(t - +U(t - 7) - R,U(t - 7))

I * z4P(t - r), R,U(t - 7)).

Now we define

iF(x,t) = u(x,t) - U”(x,t), P(s,t) =p(x,t) - P”(x,t).

Thus, we have from (5.1) and (3.2) that

(v,(t),Ey + (gd(a(t), C(t) + Cl”(t)) + ;d(uyt), C(t))

1 - Z”(ti(t - ~),ti(t - T) + u”(t - T)) .

Spectral Approximation of Fhid Flow

1 - 2d(uN(t - T), ti(t - r)),@

+(+~(t),W) + l@?yt),vw)

= (

c Ej,~~ - j= 1.2.3.8 1

(VE,,lv) - v(VE5,V\T)

3 (e(t),w) + gd(P(t),ci(t) + v”(t)) +

( ;d(P(t).ri(t))

1 -:d

( P(t - r),ti(t - sx(t - T))

1 -zd(P’(t - r),ti(t - r)),\V = -

1 ( c Ej,w

j=S.i.9

17(O) = 0 F(*) = 0,

U(T) = P” U(0) + (

T:(o) - (I(r)).

P(7) = P” P(0) + T- (

;; (0) - P(T)) l

By an argument similar to that in the previous section, we obtain

(IIW12 +IImlle)t + (5 + l b(llir,(~f +Il~~f,ll~)

- m(l(Q(t - T)(IP +II,P;(t - T,II’)

+ 2v(l - E)lti(t)l; + ;(l + a)(lir,(t)l:)l

d A,(IIti(t)f +I&t)Il’ +llti(t - T)II’

+I)P(t - T)II”) + B(llti(t - T)I12 +IIqt)ll”)

145

146 A. RASHID, C. WEIMING, AND C. BENYU

+IlF(t - &) + G,(t), (5.2)

where

4 =l+qHa) EV

cllrN2

+ -I+ -(l + 2) 819 cps 111 1.x + c 89 E

I -

+ “+ 111 PS Ill 1.x + (’ ev

G,(f) c I j=1.3.3.6.~,F.9

Spedral Approximation of Fluid Flow

t--7 + T c G,(P).

t*=7

Similarly, we have

E,(t) G P*Wexpww),

provided t E S,. t < t, , and

147

24

P*(t) $ - B = c7- IN-n-P*

Thus, in order to get the rate of convergence, we need only to estimate p,(t). Assume nz > 12/e and that U, P, UO, PO, and f are suitably smooth. Then

t’=r

Ty Ipl(tylj.2 < CT

t’=t

7 y 11 q t’) 11” < CT

t’=r

X

t-T

7 c II Ed( t’) 11” G c~“llPll2H”(o.T, L”), t’=r

t’=T

148 A. BASMID, C. WEIMING, AND G. BENYTJ

Consequen+_ we have p,(t) = O(# + N-‘“), and so obtain the fonowing

re?mh.

THEOREM 5.1. Assume T = O(N-“) with S > 2, m > n/2 ad that Y E C”(0, T; H”*’ ) gl H3(0, T; L?) n H”(0, T; L= n H”), P E C”(0, T; H”) 68 H”(O, T; L” n H ‘) n H3(0, T; L’?), and f E h “(0, T; L?. lf condition (5.3) is f;lljZed, then there exists a constant M2 depending only on v, and the nom of I$, P, and f in the spaces mentioned abme, such that for ail t E Sr9 t < t,,

REFERENCES

P. J. Roache, Computational Fluid Dynamics, 2nd ed., Hermosa Publishers, Albuquerque, 1976.

C. Canuto, _M. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Fluid Dynamics, Springer-Verlag, 1987.

Guo Benyu and Cao Weiming, Spectral method for fluid flow with low Mach number, Chinese Sci. Bull. 31:1081-1086 (1986). Kuo Penyu, Difference methods for fluid dynamics, IV. Nwner. Math. J. Chinese Unit;. 2:64-R (1990).

Spectral Approhuztion of Fluid Flow 149

A. Rashid, Cao Weiming, and Guo Benyu, The Fourier pseudospectral method for fluid flow with low Mach number, submitted for publication, 1992. Ma He-ping and Guo Benyu, The Fourier pseudospectral method with a restraint operator for the KDV equation, J. Comput. Phys. 65:120-137 (1986). Z. Zlater, R. Berkowicez, and L. P. P&m, Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants, J. Comput. Phys. 55278-301 (1984). A. Zakaria, Etude de Divers Schemas Pseudo-Spectraux de Typ Collocation pour la Resolution des Equation aux Derivees Particalles Application aux Equations de Navier-Stokes, Thesis, Univ. of Nice, 1985. H. Tal-Ezer, Spectral method in time for par&&c problems, SZm1.J. Namer. Anal. 26: 1 - 11 (1989).