Hyperbolic Initial-Boundary Value Problems - American Mathematical

MATHEMATICS OF COMPUTATION VOLUME 53, NUMBER 188 OCTOBER 1989, PAGES 547-561

Stability and Convergence of Spectral Methods for Hyperbolic Initial-Boundary Value Problems By P. Dutt Abstract. In this paper we present a modified version of the pseudospectral method for solving initial-boundary value systems of hyperbolic partial differential equations. We are able to avoid problems of instability by regularizing the boundary conditions. We prove the stability and convergence of our proposed scheme and obtain error estimates.

1. Introduction. In this paper we study the stability and convergence of spectral methods for the approximation of initial-boundary value hyperbolic systems with constant coefficients. This problem has been studied by Gottlieb, Lustman and Tadmor [1], [2] under the assumption that the boundary conditions are dissipative. We prove that a modified version of the numerical scheme they have proposed is stable and converges to the true solution of the hyperbolic initial-boundary value problem (IBVP), without any assumption of dissipativity on the boundary conditions. Our treatment closely follows their approach. Definitions. Consider the first-order hyperbolic system of partial differential equations

du _ dt

(1.1a)

du dx '

Kx
0.

Here, u = u(x, t) = (u1,..., un)T is the vector of unknowns, and A is a fixed nxn coefficient matrix. Since by hyperbolicity, A is similar to a real diagonal matrix, we may assume without loss of generality that it is diagonal:

«i

(1.1b) A

A1

0

ai+i

0

< 0,

An

>0.

A11 =

oo. Our proof relies on the stability of the modified spectral approximation (1.3) which we establish in Section 2.

In another paper we shall present the results obtained from implementing the numerical scheme we propose in this paper, and also examine the optimal choice of the approximate identity employed in the method.

2. Stability of the Modified Spectral Method.

In discussing the stability

of the numerical scheme, a closely related question concerns the conditions under

which the hyperbolic IBVP itself is stable. To establish the well-posedness of (1.1), we must establish the following inequality with some r)o > 0: roo

(2.1)

n

Jo

roo

e-a,"||u(x,i)l|2dt

< const /

e-2rit\g(t)\2 dt

Jo

for all n > r)o- Here,

iiu(x,i)n2=y

iu(M)i2max{0,(log(|L|-|Ä|)-|i4|)/4}. We now return

to the stability

of the modified spectral

approximation

(1.3).

Definition. The approximation (1.3a)-(1.3c) is stable if there exists a weighting pair u(x) = (uI(x),ulï(x)) and constants a and r/o, and an integer TV0,such that for all r¡ > r¡o and TV> N0 we have roo

(2.2a)

r¡

Jo

roo

e'2^\\vN(x,

t)\\2 dt < const N2a

e~2rit\g(t)\2 dt,

Jo

where :|v,v(x,i)||2

= \\yN(x,t)\\l

= ¡

\vlN(x,t)\2u1(x)dx

(2.2b) + Í

\v%(x,t)\2uu(x)dx.

For spectral methods using Chebyshev polynomials, we choose ul(x) = (l-x)/(l-x2)1/2

and

uu(x)

= (1 + x)/(l

- x2)1'2.

We now state the main theorem of this section. THEOREM 1.

The modified pseudospectral approximation

(1.3) is stable.

To prove this, we first look at the solution of the scalar problem /„ ,v (2.3)

dpN dpN —=a—+a(t)TN+1(x)

,

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STABILITY AND CONVERGENCE OF SPECTRAL METHODS

and zero initial data. We need to introduce some notation. h(s) denote the Laplace transform of h(t),

551

Let s = n + iÇ. Let

/»oo

(2.4)

h(s)=

\

Jo

e~sth(t)dt.

We have assumed in this definition that

h(t) =0

for t < 0.

Taking the Laplace transform of (2.3), we get spN(x, s) = a—pN(x,s)

+ â(s)T'N+1(x).

This leads us to Theorem 5.2 of [1]: THEOREM 5.2.

Let plN(x,s) be a polynomial in x of degree TVwhich satisfies

the scalar inflow problem

(2.5a)

splN(x,s) = a—plN(x,s)

+ T'N+1(x),

a < 0.

7/p5v(-1,so) =0, then Res0 < 0. In fact, plN(x,s) satisfies

(2.5b)

|&(M)I 0. Similarly, in the outflow scalar case:

(2.6a) î/p"(l,so)

sp%(x,s)=a—p%(x,s)+T'N+1(x),

a > 0,

= 0, then Reso < 0, and we have

(2.6b)

|$(M)I>I$(-M)I

for s such that Re s = r¡ > 0. Now we take the Laplace transform of (1.3) with respect to t to obtain svjv = ¿—-h

(2.7)

vîv(-l,S)

f(s)T'N+1(x),

= ^r^yv^(-l,S)

+ gI(S),

vIJv(l,«)= 7jf^)V]v(l,S)+gII(s), since the Laplace transform F£(s) of F£(i) is

&(«) = 1/(1 + M). Let Pjf(x,s) trices

= pJyfjE,s, .A1) and P$(x,s)

= p^,(x,s,Au)

denote the diagonal ma-

N

(2.8a)

Ph(x,s) = Y^k-1(T'N+1(x))^(Aï)k, fc=0

(2.8b)

P$(x,s) = Ys-k-1(TN+i(*)){k)(¿U)kfc=0


P. DUTT

552

Then the solution of (2.7) is given by (2.9)

v^x,

s) = PlN(x, s)fl,

v%(x, s) = P%(x, s)t11,

where f1 and fn satisfy the linear system

-LP$(-l,s) (l + es)

(2.10)

-RPhihs)

t\s) i"(s)

P#(M)

gu(«)

(l + es) Let 6 be a positive constant less than one. Choose

%>max{0,(log(|¿|.|A|/(l-«))-|A|/4)}. LEMMA 1.

There exists an integer TV0(e) such that for all s with Res = n > rjo

and all TV> TVo(e)there holds detDN(s) ^0, where

-LP#(-l,s) (l + es)

DN(s) =

(2.11)

-RPk(l,s)

P%(1,8)

(l + es) Here,

(2.12)

-LP$(-l,s (l + es)Pj}(l,s)

DN(s) =

-RP'Ahs) -l,s) L(l + es)PL(-l,s)

^(~M)

I

0

0

P%(h*)

= EN(s) ■PN(s). To prove the lemma, we examine the behavior of the family of functions

Me)=-íw, as TV—► oo. (Here and below, the ratio of two diagonal matrices is meant to be the product of the numerator matrix and the inverse of the denominator matrix.)

This leads us to LEMMA 2. The family of diagonal matrices (¡>n(s) converge uniformly on every compact subset of Re s > 0 to 0(s) = e-^")-1S.

Proof. By Theorem 5.2 of [1] we have that 4>n(s) is analytic and | 0, for all TV. Hence { 0 to 6(s)=e2(A'^s.


554

P. DUTT

Now by (2.12), DN(s) = EN(s) ■PN(s).

By Theorem 5.2 of [1] we know that Pn(s) is invertible; so to prove that det D^(s) t¿ 0 we need to show that Em(s) is invertible. We may write En(s) as EN(s) =

I -mN

—Sn i

where a,

i n

L(pN(s)

^n(s)

= j-— and (l U + es) Choose d so large that for all s with |s| >dwe

|L|-|ß|/|l

RÔn(s)

?n{s) = v; (l + es)'

have

+ es|2 r)o and |s| < d}. Then by Lemma 2, n(s)■On (s) converges uniformly on K to

Hence, we can find an integer TV0such that for all TV> TV0

\4>n(s)\-\0n(s)\
0 and all TV > TV0(e). This completes the proof of Lemma

1. □ We can now prove Theorem 1. System (1.3) consists of / inflow equations

d = an—vl+T]T', '"-"'dx"»

,s /;,) '■

1 < j < I,

[x),

JV-l

and (n —I) outflow equations su JN = a)^N

+ t3T'N+1(x),

l + l rjo we have for arbitrary vectors r

v\K(x,s)\\l

< constTV2V„(-1,S)|2,

v\K(x,s)\\ln

< const N2a\v3N(l,s)\2,

1 < j < I, l + l!v(-1,8) 'N

Since by Lemma 2, E^(s)

o P»(1,S)

= E-Nl(s)

1,S)

k\s)

Le»

has a uniformly bounded inverse for Res = r? > no and

TV> TV0,we get \vlN(-l,s)\2

+ \y^(l,s)\2 r¡o- We sketch the proof. Taking the Laplace transform of (3.3), we obtain

Í-AÍ,dx (3.5)

îI(-l,s)

= ^^î"(-l,s)

+ gI(s),

ÎII(1's)=(ïT^ÎI(1's)

+ êII(s)-

The solution of (3.5) is îI(x,s)=es^,)"1(*+1>AI(s),

3.6

K

'

n(x,s)=e^")

„

,

W

_1(x-i)An(s)


dt

STABILITY AND CONVERGENCE

where A(s) = (AI(s), An(s))r

OF SPECTRAL METHODS

557

satisfies the linear system

X(s)A(s) = g(s). Here, _Le-23(A»)-'

(l + es)

X(s) =

-ReMA'y I (l + es) It is easy to show, as in Section 2, that for Res = r¡ > 770the inverse X_1(s) exists and satisfies the inequality

\X-\s)\ r¡oPutting together the relations (3.2), (3.10) and (3.11), we obtain the main estimate of this section: rOO

n

||u(x,í)-w(x,í,e)||2e-2"tíTí

./o

(3.12)

< const-e2I Y \ I

\°x{(x)\2dx + r

\a noIn particular,

we have that rOO

(3.13)

T] /

||u(x,í)-w(x,í,e)||2e-2,"dí

= 0(e2)

for r\ > T]0.

4. Convergence of the Modified Spectral Approximation. In the preceding section we proved that w(x,i,e), the solution of the modified IBVP (1.4), converges to u(x, f ) linearly in e as e -+ 0. Hence, to prove that v^(x,t,e), the solution of the modified spectral approximation (1.3), converges to u(x,i) as e —+0 and TV—»oo, it is enough to show that for fixed e, v;v(x,i,e) —+w(x, i,e) as TV—► oo. The proof of this relies on the stability result we have established in Section 2, and is essentially the same as the proof of convergence in [2]. However, for the sake of completeness we shall provide a sketch of the proof.


STABILITY AND CONVERGENCE

OF SPECTRAL

METHODS

559

Let r be the solution of

dr _ dr dt dx' r(x, 0) = f (x) = w(x, 0, e),

(4.1)

rI(-l,í)=wI(-l,í,e), rII(l,í)

= wII(l,í,e).

Then r(x, i) = w(x, t, e). Let s be the pseudospectral

approximation,

ds

,ds

„,.

, . ., .

dï=Ao-x+T»^X)-m> (4.2)

a(x, 0) = Eî (x) = Ew(x, 0, e), sI(-l,í)

= wI(-l,í,e),

sII(l,í)

= wII(l,í,e).

Here, we define the projection operator E = (El,Eu) by requiring that for any function F, ElF and EnF are polynomials of degree TVat most, satisfying

EIF(x]) = F(x3), EuF(xJ) = F(x]),

j = l,...,N + l, j = 0,...,N,

where the points Xj are defined by

Xj = cos(ttj/(N + 1)),

j = 0,...,N

+ l.

Then we define

Ef=(E*f\Eufu)T. Let 6 — s —Er and a = Er —r. Then it is shown in [2] that for n > 0 the following

estimate holds: /•OO

r,J

i

roo

e-2«t\\6(x,t)\\ldt + — jo e-2^\6\-l,t)\2 1

(4.3)

dt

f°°

+—jo

e-2^\6l\l,t)\2

dt

roo

< const/

Jo

e-2rit\\\Q(x,t)\\\2 dt,

where Q = (Q\QU)T is defined by

Q' = A*(e^ \

ox

- ^-EW) , ox

j

Q» = A- (e^ \

ox

and N

\\\Q(x,t)\\\2= ^Y^-x^il

+ x^1^^

3=1

+ ^E(1

+ ^)2(1-^)i«II(^'i)i2-

3= 1


- ^\") ox

)

,

P. DUTT

560

Next, we compare the modified spectral solution s with v, as defined by (1.3). Then v —s satisfies |(v_8)

(4.4)

= |_(v_9)+T;+i(x).(r_ö);

(v-s)(x,0)

= 0,

(v1 - s'X-l, t) = L(((v" - s»)(-l) * Fe)(t)) + bHt), (v" - s")(l, t) = R(((vl - 8^(1) * FE)(t)) + b"(i).

where

b1(t) = L(((6u + vu)(-l)*Fe)(t)),

bn{t) = R{{{S1+ a1){l)*Fe){t)). By Theorem 1 of Section 2 we then have roo

(4.5)

r) /

Jo

rOO

e-2"*\\a -v||2di