NASA Contnetor Report 178063

lCASE REPORT NO. 86-8

leASE

NASA-CR-1780G3

19860014826

CONVERGENCE OF SPECTRAL METHODS FOR HYPERBOLIC

INITIAL-BOUNDARY VALUE SYSTEMS

David Gottlieb

Liviu Lustman

Eitan Tadmor

Contract Nos. NASl-17070 and NASl-lSl07

January 1986

INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING

NASA Langley Research Center, Hampton, Virginia 23665

Operated by the Universities Space Research Association

NI\SI\

Nahonal AeronautiCS and

Space Administration

Langley Research center

Hampton.Virginia 23665

UJ~GLEY f(f::S~ARCH CENTER

Ll3::?J,RY, NASA

!~.'\~.~:'TCN, VIR81NIA

CONVERGENCEOF SPECTRALMETHODS

FOR HYPERBOLIC INITIAL-BOUNDARY VALUE SYSTEMS

David Gottlieb (1'2'3)

Liviu Lustman (2)

Eitan Tadmor (1,2,4)

Abstract

We present here a convergence proof for spectral approximations for

hyperbolic systems with initial and boundary conditions. We treat in detail

Chebyshev collocation, but the final result is readily applicable to other

spectral methods, such as Legendre collocation or tau-methods.

(I) School of Mathematical Sciences, Tel-Aviv University

(2) Institute for Computer Applications in Science and Engineering

(3) Division of Applied Mathematics, Brown University

(4) Bat-Sheva Foundation Fellow

Research was supported in part by the National Aeronautics and Space

Administration under NASA Contract No. NASI-17070 and NAS1-18107 while the

authors were in residence at ICASE, NASA Langley Research Center, Hampton,

VA 23665. Additional support was also provided in part by AFOSR 85-0303 for

the first author, and by NSF Grant No. DMS 85-03294 and ARO Grant No. DAAG29-

85-K-0190 for the third author while in residence at UCLA, Los Angeles, CA

90024.

INTRODUCTION

In the paper [I], we derived stability results for spectral methods

applied to inltial-boundary value problems for hyperbolic systems. The paper

demonstrates that one can bound certain weighted L 2 spatial norms of the

solution in terms of norms of the boundary data (homogenous initial conditions

are assumed). The bounds also contain powers of N, which is the degree of

the approximating polynomials.

Here we show that the approximations discussed above actually converge to

the exact solution, at least when this solution is smooth. We bound the error

in the numerical method by a power of N multiplying a term which depends

only on the exact solutlon--more precisely, this is the interpolation error of

the initial value and boundary derivatives. For sufficiently differentiable

functions, this interpolation error will decay fast enough to drive the full

approximation error to zero. We have not attempted to derive the sharpest

bound of this type, but merely to show that such a bound exists.

The method of proof here is similar to the one in [1,2], where basic

results are first deduced for a scalar equation, and then extended to the full

system. Accordingly, the paper is divided into two sections, the first

dealing with the scalar case and the second with the system. By the means of

Gauss-Lobatto quadrature formulas, we first bound the error at outflow for a

single scalar equation. Then we use this estimate, together with the basic

stability result of [i], to bound the overall error for a system.
-2-

1. THE SCALAR CASE - CHEBYSHEV COLLOCATION

Given the equation

ut =ux Ixf< 1,t > 0

(1.1) u(x,0) = f(x)

u(l,t) = g(t),

we consider the pseudospectral method based on collocating at the extrema of

TN+I, where Tm is the Chebyshev polynomial of degree m

Tm(X) = cos(m eos-l(x)).

It is shown in [3,4] that the pseudospectral approximation v = VN(X,t )

satisfies

8v _v

8t - 8x + _(t) T_+I(X)

(1.2) v(x,0) = PN f

v(l,t) = g(t),

with the projection operator PN defined by

PN f is a polynomial of degree < N

(1.3)

(PN _J f)(x) = f(x) at x = xj = cos _ , 0 < j < N.
-3-

Define now 6N(X,t) as the difference between vN and the projection of u

6N(x,t) = v - PN u.

This function satisfies the equation:

_-_ _N(X, t) = _-_ _ _N (x't) - [PN _ux _x PN u] + T TN+ I

(1.4) 'iI _N(X,0) = 0

_N(l,t) = 0.

We note at this stage that the polynomial (l+x)6 N T_+ 1 is of degree 2N+I

and therefore may be integrated exactly by the Gauss-Lobatto quadrature

rule. This results in the following

N+I 1 (l+x)6 N TN+ 1 _ 1

f dx = _ j_0 T. (I + xj) _N(Xj) TN+I(X j) = 0, -I ¢ l_x 2 "= 3

(1.5)

co = CN+ 1 = 2, cj = 1 for 0 < j _< N,

since i + XN+ I = 0, 6N(X 0) = 0 and T_+l(X j) = 0 for j = I,..-,N. In

fact, because of the term (l-x) and the boundary condition, the indices in

the quadrature sums may run only from i to N (instead of 0 to N+I).

Defining

_u

QN - _x (PN u) - PN _-x '

multiply (1.4) by (I + x)_ N and integrate to get
-4-

1 d 1 (l+x)6_ (x,t) 1 (l+X)6N(X,t)

2 dt f dx = f _--_6N(X,t)dx

-i _ 1 - x2 -I _ 1 - x2

(1.6)

1 (l+X)6N(X,t)Q N

+ _ dx.

-I J 1 - x 2

An integration by parts on the right-hand side produces:

I (l+x)62(x,t)dx 1 _2(x,t)

1 d i dx= 1 i dx

d-{-I / 1 - x 2 - _-I (l-x) / 1 - x2

(i.7)

1 (I+X)6N(X,t)QN(X,t)

+ _ dx.

-I / 1 - x2

We again use the Gauss-Lobatto formula and reach

1 d 1 (l+x)6_(x,t)

2 dt f dx

-I _ 1 - x 2

N 62(xj ,t) N

= --- _ _-i t) + _ 7 (I + )6N(Xj,t)QN(x j t) 2N j=lI 1-x3 8N6N ' Nj=I xj

(i.8)

N 62(xj 62(x i

IT ,t) 2 N ,t)

- 2-_ l 1 - x 8N 6N (-l,t) +-i-N [" 1 - x. • _ • . 31 3 31 3

N

)2 2(xj

+_-_ Z (I + x. (I - x )Q ,t).

.] 1 .l j
-5-

Equation (1.8) yields immedlately--slnce _N(X,0) = 0 --

i (l+x) _ t t

(1.9) S _ 62(x't)dx +TN _ 62(-l't)dt < _ IIIQN (x't)lll2 dt

-I Jl_x 2 0 0

where

N

(I.I0) lllQ(x,t)lll 2 = _-N I (l+xj)2 (l-x.)Q2(xj ,t)'J

j=l

In the next section we will need a different version of (1.9), in which

-2nt

the time integral is weighted by e , n > 0:

(1.11) S e-2nt S (l+x)_(x,t)dxdt + 8--_- S e-2_t 6_(-1,t)dt

0 -I _ l_x2 0

I e-2_t 2

< _ I IIIQN(X,t)II I dt. 0

This form matches the Laplace-Fourier transforms which are used in the basic

stability estimates.

It should be noted that the bound on 6_('l,t) obtained in (1.11)

(1.12) I e-2nt 62(-l,t)dt < =(N) S e-2nt IIIQN(X,t)lll 2 dt, 0 0

4N

e(N) = --

is very crude. A better estimate can be found for the boundary error u(-l,t)

- v(-l,t) by taking the Laplace transform of (i.I) and (1.2) and analyzing

the difference. This approach was used by Dubiner [5] who obtained
-6-

~ 1

(u-v)(-1,t) ~ _. The same bound holds for PN u - v, since, for smooth

1

functions, PN u approaches u faster than any power of _ .

2. CONVERGENCE PROOF FOR SYSTEMS

Consider the first-order hyperbolic system of partial differential

equations

(2.1) 3u _ A 3u _t _-_, Ixl< 1,t>0

where

u = u(x,t) = (u(1)(x,t), u(2)(x,t),...,u(n)(x,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, without loss of

generality, take it diagonal:

I )

0

A=

0 A ll

(2.2)

A I = •". <0, All = •• >0.

a£ "an

The solution of this system is uniquely determined if we specify initial

conditions
-7-

(2.3) u(x,0) = f(x) = (fl(x), fll(x))

and boundary conditions

ul(-1,t) = Lull(-1,t) + g I(t)

(2.4)

ull(l,t) = Rul(l,t) + gll(t).

In these formulas, f and g = g(t) = (gl(t), gll(t)) are prescribed nvectors, and

(uI, uIl)

(2.5) (fl, fll)

igl,gii)

is the partition of these vectors into inflow and outflow components--

corresponding to the partition of A in (2.1b). L and R are constant

reflection matrices of order _ x (n-_) and (n-_) x _, respectively.

We shall discuss only problems whose solutions decay in time and

therefore postulate:

ASSUMPTION I. There exists a constant y > 0, such that

IRl ILl < I - y < I.
-8-

(IAI and Ivl denote the Euclidean norm of a matrix A or vector v,

respectively. )

In a pseudospectral Chebyshev approximation to (2.1), one seeks a vector

v = vN of N-degree polynomials such that

(2.6a) _v A _v .

_--_= -_ + T(t) TN+I(X).

The multiplier r = (TI, TII) is determined by the boundary conditions

vl(-l,t) = Lvll(-l,t) + gl(t)

(2.6.b)

vll(l,t) = Rvl(l,t) + gll(t).

Initially, v is defined by collocation:

I vl(x,O) = fl[x) at x = x., j = 1,2,---,N+I

P %

3

(2.6c)

vll(x,0) = fll(x) at x = x., j = 0,1,-..,N. 3

In [I] the stability of the approximation (2.6) has been established, under

zero initial conditions. Here we shall prove the convergence of VN(X,t )

to u(x,t) as N tends to infinity.

We define a pair of projection operators p= (pl, pll) by requiring that

for any function F, pl F and pll F be polynomials of degree N at most,

satisfying:
-9-

(pl F)(xj) = F(xj), j = 1,2,...,N+I

(2.7)

(pll F)(xj) = F(xj), j = 0,1,...,N.

Note that two distinct sets of nodes are used for collocation - cf. (2.6c).

We can state now the main convergence result:

THEOREM. Let g = SN(X,t) = v-Pu be the error in the pseudospectral

approximation (2.6) to the hyperbolic system (I.1). Let Q = (QI, QII) be

the approximation error

QI = Al(pl auaxI aax pl u I)

(2.8)

QII = A£1(pll auII aax pll ull) "

Then

oo oo

(2.9) f e-2qt lls(x,t);;2 dt < KNa(N) e-2nt 2

0 2_ f0 lllQ(x,t)lll at

with K independent of N, a(N) as defined in (1.12) and
-i0-

(2.10)

1 1

llc(x,t)l'2= / (l-x)I_l(x,t)l 2 dx + / (1+x)]€ll(x,t)l2 dx

-I / 1 - x 2 -I / i - x2

N

IIIQ(x,t)ll]- _ Y (l-x.)2(l+x.) IQl(xj,t)l 2 N j=l 3 3

N

+ N_j=l_ (l+xj)2 (l-xj) IQll(xj,t)l2

(Here we generalize the seminorm defined in (i.II).)

Proof: Let r be the solution of

_r 8r

_t _x

r(x,O) = u(x,O)

(2.11)

rI(-1,t) = uI(-1,t)

rlI(1,t) = ulI(1,t),

and let s be the pseudospectral Chebyshev approximation to (2.11), i.e.:

8s = A _s

8--{ _-_x+ T_+I(X) 0(t)

s(x,O) = Pu(x,O)

(2.12)

sI(-1,t) = uI(-1,t)

sII(1,t) = ulI(1,t).
-11-

It is obvious that r _ u, but s does not satisfy the boundary conditions

(2.6c) and therefore its multiplier e is distinct from T. In any case, for

= s - Pr, we have:

(2.13)

OO CO

I e-2qt 61 2

f e-2_t ll6(x,t)ll2 dt +2-N_n f I (l,t) I dt 0 0

0o

1 e-2q t

+ _- _ 1611(_l,t)idt _ 1 _ e-mnt lllQ(x,t)lll 2 at. 0 n 0

This is clearly a restatement of (i.Ii). Next, we compare the spectral

solution s with v -- as defined by (2.6). We can show that

oo Go

(2.14) _ e-2qt lls-vl2, dt < --cN _ e-2nt[161(l,t)12 + [_ll(-l,t)12]dt.

0 n 0

Indeed, v-s satisfies:

a (v-s) a a--{ =_-_ (v-s) + r_+l(T-e)

(v-s)(x,0)= 0

_ _ I (vI sl)(-l,t) = L(vII sll)(-l;t) + Lsll(-l,t) - ul(-l,t) + g

= L(v II - sll)(-l,t) + L611(-l,t),

(vII - sII)(-l,t)= R(v[ - sI)(1,t)+ R6I(1,t),
-12-

and the inequality (2.14) follows from the stability estimate in [I, Theorem

5.1].

Now, since Pr = Pu we have

g - _ = V-S,

and, hence,

_ CO

/ e-2at lls(x,t)i2l dt < / e-2_t ]16(x,t);l2 dt + / e-2_t llv-sll2 dt.

0 0 0

The first term is majorized in (2.13) and the second in (2.14), thus

establishing the theorem.

Again, we emphasize that the bound for u - vN is the same as the bound

for Pu - vN --which we just have computed--for smooth functions u, which

Pu approaches rapidly.

We conclude with two remarks:

a. The stability estimate of (2.14) explicitly uses boundary values.

This is why these have to be bounded beforehand, which was done in the

preceding section, formula (I.I0).

b. Our result applies not only to Chebyshev collocation, but also to

other spectral methods (which, however, have to satisfy assumptions I, II and

III of [I]). Indeed, the only quantity that varies with the spectral method

employed is the coefficient weight at ±I for the Gauss-Lobatto quadrature.

~ I

Once it is known that this weight is .... which is the case for Gegenbauer

Nk

collocation, as shown in [1]--the same proof follows through. In particular,

it is sufficient to evaluate outflow errors for a scalar equation in order to

estimate errors at both boundary points for a system.
-13-

_FE_N_S

[i] D. GOTTLIEB, L. LUSTMAN, and E. TADMOR, Stability analysis of spectral

methods for hyperbolic initial boundary values systems, NASA Contractor

Report No. 178035, ICASE Report No. 86-2.

[2] L. LUSTMAN, The time evolution of spectral discretizations of hyperbolic

systems, NASA Contractor Report No. 172432, ICASE Report No. 84-37,

SINUM, to appear.

[3] D. GOTTLIEB and S. A. ORSZAG, Numerical Analysis of Spectral Methods:

Theory and Application, SIAM, Philadelphia, 1977.

[4] R. G. VOIGT, D. GOTTLIEB, and M. Y. HUSSAINI, (eds.), Spectral Methods

for Partial Differential Equations, SIAM, Philadelphia, 1984.

[5] M. DUBINER, Ph.D. Thesis, Department of Applied Mathematics,

Massachusetts Institute of Technology, 1981.
1. Report No. NASA CR-178063 12. Government Accession No. 3. Recipient's Catalog No.

ICASE Report No. 86-8

4. Title and Subtitle 5. Report Date

CONVERGENCE OF SPECTRAL METHODS FOR January 1986

HYPERBOLIC INITIAL-BOUNDARY VALUE SYSTEMS 6. Performing Organization Code

7. Author(s) 8.86

E!<gming Organization Report No. David Gottlieb, Liviu Lustman, and Eitan Tadmor

9. Performing Organization Name and Address 10. Work Unit No.

Institute for Computer Applications in Science

and Engineering 11. Contract or Grant No.

Mail Stop l32C, NASA Langley Research Center NASl-17070; NASl-18107 u VA ?1r,r,') .')??')

12. Spo~soring Agency Name and Address 13. Type of Report and Period Covered

Contractor Report

National Aeronautics and Space Administration

Washington, D.C. 20546 14. Sponsoring Agency Code

505-31-83-01

15. Supplementary Notes

Langley Technical Monitor: Submitted to SIAM J. Numer. Anal.

J. C. South

Final Report

16. Abstract

We present here a convergence proof for spectral approximations for

hyperbolic systems with initial and boundary conditions. We treat in detail

Chebyshev collocation, but the final result is readily applicable to other

spectral methods, such as Legendre collocation or tau-methods.

17. Key Words (Suggested by Authors(s» 18. Distribution Statement

spectral methods 64 - Numerical Analysis

hyperbolic systems

convergence

stability Unclassified - unlimited

19. fl~~y.~s~rtHoJ this report) \20tJsecfrity Clr:f.if.wf this page) 21. No. of Pages 122. Price

nc ass! e 15 A02

For sale by the National Technical Information Service, Springfield, Virginia 22161

NASA-Langley, 1986