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TW 2006_01

Jan Van lent
Multigrid methods for time-dependent partial differential equations

Advisor(s): Stefan Vandewalle

Abstract

Time-dependent partial differential equations are solved numerically by discretizing both space and time. Since the resulting systems of equations can be very large, it is often necessary to use iterative methods that exploit the structure of these systems. For discretized parabolic problems multigrid methods are a particularly good choice. A typical model problem is the heat equation, discretized using finite differences or finite elements in space and a linear multistep method in time. We investigate here how multigrid techniques can be used for more general time-dependent problems. In particular we develop multigrid methods for anisotropic problems, high order time discretizations and problems with delay. Furthermore, we propose a new framework for the convergence analysis of multigrid methods for time-dependent partial differences equations.

Anisotropic partial differential equations have coefficients with a strong directional dependency. For such problems standard multigrid methods break down. By extending the techniques for stationary anisotropic problems, we develop efficient multigrid methods for time-dependent anisotropic problems. We consider methods based on line relaxation, semicoarsening and multiple semicoarsening. The same methods are also applied with good results to diffusion equations with coefficients that depend on position as well as direction.

Implicit Runge-Kutta methods, boundary value methods and general linear methods are powerful time discretization schemes providing high order accuracy, good stability and many other desirable properties. For general time-dependent problems, however, the resulting systems of equations are harder to solve than the ones for linear multistep methods. We show that for discretized parabolic problems, very efficient multigrid methods can be developed. The stability of the time discretization schemes turns out to be very important for the convergence of the iterative methods. The same techniques are used to study iterative methods in combination with Chebyshev spectral collocation in time.

For standard time-dependent partial differential equations, the change of state at a certain time only depends on the current state of the system. For delay partial differential equations, the change of state also depends on the state of the system at times in the past. We study iterative methods for diffusion equations with one extra term with a fixed delay.

In all these cases the performance of the methods is assessed with a theoretical convergence analysis and numerical experiments. The theoretical analyses combine the theory of Volterra convolution operators and Laplace transforms for time-dependent problems and Fourier mode techniques for multigrid. We propose a new approach for the spectral analysis of iterative methods based on functional calculus. This theory unifies the Laplace analysis for time-dependent problems and the Fourier analysis of multigrid methods.