Overlapping Schwarz Waveform Relaxation for Parabolic Problems

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Overlapping Schwarz Waveform Relaxation for Parabolic Problems

GANDER, Martin Jakob

GANDER, Martin Jakob. Overlapping Schwarz Waveform Relaxation for Parabolic Problems. In:

10th International Conference on Domain Decomposition Methods. 1998. p. 425-431

Available at:

http://archive-ouverte.unige.ch/unige:8287

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Overlapping Schwarz Waveform Relaxation for Parabolic Problems

Martin J. Gander December 7, 1997

1 Introduction

We analyze a new domain decomposition algorithm to solve parabolic partial dierential equations. Two classical approaches can be found in the litera- ture:

1. Discretizing time and applying domain decomposition to the obtained sequence of elliptic problems, like in Kuz90], Meu91], Cai91] and references therein.

2. Discretizing space and applying waveform relaxation to the large sys- tem of ordinary dierential equations, like in LO87], JP95] and JV96]

and references therein.

In contrary to the classical approaches, we formulate a parallel solution al- gorithm without any discretization. We decompose the domain into over- lapping subdomains in space and we consider the parabolic problem on each subdomain over a given time interval. We solve iteratively parabolic prob- lems on subdomains, exchanging boundary information at the interfaces of subdomains. This algorithm is like a classical overlapping additive Schwarz, but on subdomains, a time dependent problem is solved, like in waveform relaxation. Thus the name overlapping Schwarz waveform relaxation.

Scientic Computing and Computational Mathematics, Stanford University, mgan- der@sccm.stanford.edu

1

t

0

^{1 1 2 2}

L

L L x

Figure 1: Decomposition into two overlapping subdomains

This algorithm has been considered before in GS97], where linear conver- gence on unbounded time intervals was proved for the one dimensional heat equation, and in GK97] where superlinear convergence for bounded time intervals was shown for a constant coecient convection diusion equation.

We study here three model problems which show that the results in GS97]

and GK97] hold for more general parabolic equations. All the convergence results are inL¹ with the norm^jju^jj1:= sup_xt^ju(xt)^j. We denote byuthe exact solution of the problem under consideration and by u^k the k-th iterate of the algorithm.

2 Reaction Diusion Equation

We study the overlapping Schwarz waveform relaxation algorithm for the one dimensional reaction diusion equation with variable diusion coecient

@u@t ⁼c²(xt)@²u

@x^{2 +}f(u) 0< x < L 0< t < T

with appropriate initial and boundary conditions. We consider only the two subdomain problem given in Figure 1 with nonempty overlap, 0< < <1.

The generalization to N subdomains can be found in Gan97a].

2.1 Theoretical Results

Dene for 0< x < L and 0< t < T

^c² := sup_xt c²(xt) ^a:= sup_xt

2IR

f⁰() c²(xt) which are assumed to be nite.

2

Theorem 2.1 (Linear Convergence for

t²0¹)

)

^{If^}a < _L² then the overlapping Schwarz waveform relaxation algorithm converges linearly on un- bounded time intervals,

jju^;u^2k+1jj1kjju^;u^0jj1 with convergence factor

= sin(

p^a(1^;)L)sin(^p^aL) sin(^p^a(1^;)L)sin(^p^aL) <1:

Proof

The proof can be found in Gan97b].

Note that the convergence factor tends to 1 and convergence is lost as the overlap goes to zero or ^a goes to (=L)². On the other hand becomes small for ^a large and negative.

Theorem 2.2 (Superlinear Convergence for

t²0T]

)

The overlapping Schwarz waveform relaxation algorithm converges superlinearly on bounded time intervals,

jju^;u^2k+1jj1max(1e^{2^c2aT^} )erfc(k(^;)L

p^c²T ^)jju^;u^0jj1:

Proof

The proof can be found in Gan97b].

Note that convergence is lost again when the overlap goes to zero (=).

But in contrary to the linear result, the superlinear convergence rate does not depend on ^a, only the constant in front does. Instead there is a dependence on the diusion coecient ^c² and the length of the time interval: the smaller the product ^c²T, the faster the convergence.

2.2 Numerical Results

We consider the model problem

@u@t ⁼ @²u

@x^{2 + 5(}u^;u³) 0< x <1 0< t < T

with initial condition u(x0) = x² and boundary conditions u(0t) = 0 and u(1t) = e^;t. We choose 20% overlap ( = 0:4 and = 0:6). On the left of Figure 2 we show the linear convergence of the algorithm for a long time intervalT = 4 and on the right of Figure 2 we show superlinear convergence of the algorithm for a short time interval T = 0:1.

3

0 1 2 3 4 5 6 7 8 9 10 10⁻³

10⁻² 10⁻¹ 10⁰

Iteration k

Error ||u−uk||

0 1 2 3 4 5 6

10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

Iteration k

Error ||u−uk||

Figure 2: Linear convergence on long time intervals on the left and superlin- ear convergence on short time intervals on the right. Dashed the numerical results and solid the derived upper bounds for the convergence rate

3 Convection Diusion Equation

We consider a convection diusion equation with variable coecients in one dimension

@u@t ⁼c²(xt)@²u

@x^{2 +}b(xt)@u

@x ⁺f(xt) 0< x < L 0< t < T with appropriate initial and boundary conditions.

3.1 Theoretical Results

Dene for 0< x < L and 0< t < T

^c² := sup_xt c²(xt) ^{^}b := sup_xt b(xt)

c²(xt) b := inf_xt b(xt) c²(xt) which are assumed to be nite.

Theorem 3.1 (Linear Convergence for

^t201)

)

The overlapping Schwarz waveform relaxation algorithm converges linearly on unbounded time inter- vals,

jju^;u^2k+1jj1kjju^;u^0jj1 4

with convergence factor

= 1^;e^;b(L;L) 1^;e^{;b(L;L) 1}

;e^{;^bL} 1^;e^{;^bL}:

Proof

The proof can be found in Gan97a].

As in the reaction diusion case convergence is lost as the overlap goes to zero. But a large ratio of convection to diusion helps the algorithm by decreasing .

Theorem 3.2 (Superlinear Convergence for

^t20T]

)

The overlapping Schwarz waveform relaxation algorithm converges superlinearly on bounded time intervals,

jju^;u^2k+1jj1e^{(;)L(^b;b)k=2} erfc(k(^;)L

p^c²T ^)jju^;u^0jj1

Proof

The proof can be found in Gan97a].

Note that the principal convergence rate is the same as in the reaction diusion case, but there is a lower order dependence on the convection term which slows the convergence down. The dependence is lower order, because erfc(x) e^;x2. If the coecients are constant, the dependence on the con- vection term disappears completely.

3.2 Numerical Results

We consider the model problem

@u@t ⁼ @²u

@x^{2 +}b@u@x ^{+ 5}e^{;(t;2)2;(x;14)2} 0< x <1 0< t < T

with the same initial and boundary conditions as before. We choose again an overlap of 20% and dierent convection terms, b ^{2 f}02:55^g. Figure 3 shows on the left the long time behavior of the algorithm for T = 3 and the three dierent values of b and on the right the short time behavior of the algorithm for T = 0:05 and b= 5.

5

0 1 2 3 4 5 6 7 8 9 10 10⁻⁶

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Iteration k

Error ||u−uk||

0 1 2 3 4 5 6

10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

Iteration k

Error ||u−uk||

Figure 3: Linear convergence on the left and superlinear convergence on the right, dashed the numerical experiment and solid the theoretical bound on the convergence rate.

4 Heat Equation in

ⁿ

-Dimensions

Consider the heat equation in n dimensions

@u@t ⁼ u+f(^xt) ^x20< t < T

with appropriate initial and boundary conditions. We assume that is a smoothly bounded domain in IRⁿ and that we have an overlapping decom- position of the domain into a nite number of subdomains GZ97].

4.1 Theoretical Results

Theorem 4.1 (Linear Convergence for

t²0¹)

)

The overlapping Schwarz waveform relaxation algorithm converges linearly on unbounded time inter- vals max_j ^jju^;u^k_j^+m+1jj1max_j ^jju^;u_kj^jj1

where < 1, dependent on the geometry but independent of k. Note that m depends on the number of subdomains.

Proof

The proof can be found in GZ97].

6

0 1

1

^{1 2}

^{3 4}

bc from ² bc from ³ bc from ⁴

Figure 4: Decomposition of the domain for the heat equation model problem

Theorem 4.2 (Superlinear Convergence for

t²0T]

)

The overlapping Schwarz waveform relaxation algorithm converges superlinearly on bounded time intervals

max_j ^jju^;u_kj^jj1 (2n)^kerfc( k

2^pnT^{)maxj jj}u^;u⁰_j^jj1 where is a parameter related to the size of the overlap.

Proof

The proof can be found in GZ97].

Note that the superlinear convergence rate does not depend on the num- ber of subdomains.

4.2 Numerical Results

We used the homogeneous heat equation in two dimensions with homoge- neous initial and boundary conditions on a unit square with the decompo- sition given in Figure 4 for our numerical experiments. Figure 5 shows for two dierent sizes of the overlap, () ^{2 f}(0:30:7)(0:40:6)^g, on the left the linear convergence behavior of the algorithm for a long time interval, T = 1 and on the right the superlinear convergence behavior for a short time interval, T = 0:01.

7

0 1 2 3 4 5 6 10⁻³

10⁻² 10⁻¹ 10⁰

Iteration k

Maximum error on the boundary

0 1 2 3 4 5 6

10⁻²⁵ 10⁻²⁰ 10⁻¹⁵ 10⁻¹⁰ 10⁻⁵ 10⁰

Iteration k

Maximum error on the boundary

Figure 5: Linear convergence on the left and superlinear convergence on the right for the two dimensional heat equation

5 Conclusion

On unbounded time intervals the proposed algorithm converges linearly, pro- vided there is a strong enough dissipation mechanism. Thus the new algo- rithm inherits the classical convergence behavior of the overlapping Schwarz algorithm for elliptic problems, and also the classical convergence behavior of waveform relaxation on long time windows.

On bounded time intervals the convergence is superlinear, which is a new result for domain decomposition algorithms. The waveform relaxation theory predicts superlinear convergence, but the presence of diusion leads to a faster superlinear convergence rate than the one predicted by waveform relaxation theory GK97].

Note that the superlinear convergence rates do not depend on the number of subdomains and thus there is no coarse mesh needed to avoid deterioration of the algorithm Gan97a].

References

Cai91] X. C. Cai. Additive Schwarz algorithms for parabolic convection- diusion equations. Numerische Mathematik, 60:41{61, 1991.

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Gan97a] M.J. Gander. Analysis of Parallel Algorithms for Time Dependent Partial Dierential Equations. PhD thesis, Stanford University, California, USA, 1997.

Gan97b] M.J. Gander. A waveform relaxation algorithm with overlapping splitting for reaction diusion equations. submitted to Numerical Linear Algebra with Applications, 1997.

GK97] E. Giladi and H.B. Keller. Space-time domain decomposition for parabolic problems. submitted to SINUM, 1997.

GS97] M.J. Gander and A.M. Stuart. Space-time continuous analysis of waveform relaxation for the heat equation. to appear in SIAM Journal on Scientic Computing, 1997.

GZ97] M.J. Gander and H. Zhao. Overlapping Schwarz waveform relax- ation for parabolic problems in higher dimension. In Proceedings of Algoritmy'97, 1997.

JP95] R. Jeltsch and B. Pohl. Waveform relaxation with overlapping splittings. SIAM J. Sci. Comput., 16:40{49, 1995.

JV96] J. Janssen and S. Vandewalle. Multigrid waveform relaxation on spatial nite-element meshes: The continuous-time case. SIAM J.

Numer. Anal., 33(6):456{474, 1996.

Kuz90] Y. Kuznetsov. Domain decomposition methods for unsteady convection-diusion problems. In Proceedings of the Ninth Interna- tional Conference in Computing Methods in Applied Sciences and Engineering, pages 211{227, 1990.

LO87] Ch. Lubich and A. Ostermann. Multi-grid dynamic iteration for parabolic equations. BIT, 27:216{234, 1987.

Meu91] G.A. Meurant. A domain decomposition method for parabolic problems. APPLIED NUMERICAL MATHEMATICS, 8:427{441, 1991.

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