Parallel Multigrid Waveform Relaxation for Parabolic Problems

Wetenschap is meer dan het object dat zij bestudeert. Wetenschap is ook de weg naar de ontdekking, en bovendien, wetenschap is ook het verhaaJ van de ontdekkingsreis. -Po Thielen Focus research, Nr 10-11, juli 1991. The numerical solution of a parabolic partial differential equation is usually calcu lated by using a time-stepping method. This precludes the efficient use of parallelism and vectorization, unless the problem to be solved at each time-level is very large. This monograph investigates the use of an algorithm that overcomes the limitations of the standard schemes by calculating the solution at many time-levels, or along a continuous time-window simultaneously. The algorithm is based on waveform relazation, a highly parallel technique for solving very large systems of ordinary differential equations, and multigrid, a very fast method for solving elliptic partial differential equations. The resulting multigrid waveform relazation method is applicable to both initial boundary value and time-periodic parabolic problems. We analyse in this book theoretical and practical aspects of the multigrid waveform relaxation algorithm. Its implementation on a distributed memory message-passing computer and its computational complexity (arithmetic complexity, communication complexity and potential for vectorization) are studied. The method has been im plemented and extensively tested on a hypercube multiprocessor with vector nodes. Results of numerical experiments are given, which illustrate a severalfold performance gain when compared to parallel implementations of a variety of standard initial bound ary value and time-periodic solvers.

Inhalt
1 Introduction.- 1.1 Numerical simulation and parallel processing.- 1.2 The simulation of time-dependent processes.- 1.3 Outline.- 2 Waveform Relaxation Methods.- 2.1 Introduction.- 2.2 Waveform relaxation: basic ideas.- 2.3 A classification of waveform methods.- 2.4 General convergence results.- 2.5 Convergence analysis for linear systems.- 2.6 Waveform relaxation acceleration techniques.- 2.7 Some concluding remarks.- 3 Waveform Relaxation Methods for Initial Boundary Value Problems.- 3.1 Introduction and notations.- 3.2 Standard waveform relaxation.- 3.3 Linear multigrid acceleration.- 3.4 Convergence analysis.- 3.5 Experimental results.- 3.6 Nonlinear multigrid waveform relaxation.- 3.7 A multigrid method on a space-time grid.- 3.8 Concluding remarks.- 4 Waveform Relaxation for Solving Time-Periodic Problems.- 4.1 Introduction.- 4.2 Standard time-periodic PDE solvers.- 4.3 Time-periodic waveform relaxation.- 4.4 Analysis of the continuous-time iteration.- 4.5 Analysis of the discrete-time iteration.- 4.6 Multigrid acceleration.- 4.7 Autonomous time-periodic problems.- 5 A Short Introduction to Parallel Computers and Parallel Computing.- 5.1 Introduction.- 5.2 Classification of parallel computers.- 5.3 The hypercube topology.- 5.4 The Intel iPSC/2 hypercube multiprocessor.- 5.5 Parallel performance parameters.- 6 Parallel Implementation of Standard Parabolic Marching Schemes.- 6.1 Introduction.- 6.2 Problem class and discretization.- 6.3 Parallel implementation: preliminaries.- 6.4 The explicit update step.- 6.5 The multigrid solver.- 6.6 The tridiagonal systems solver.- 6.7 Timing results on the Intel hypercube.- 6.8 Numerical examples.- 6.9 Concluding remarks.- 7 Computational Complexity of Multigrid Waveform Relaxation.- 7.1 Introduction.- 7.2 Arithmetic complexity.- 7.3 Parallel implementation.- 7.4 Vectorization.- 7.5 Concluding remarks.- 8 Case Studies.- 8.1 Introduction.- 8.2 Programming considerations.- 8.3 Representation of the results.- 8.4 Linear initial boundary value problems.- 8.5 Nonlinear initial boundary value problems.- 8.6 Linear time-periodic problems.- 8.7 Example 7: a nonlinear periodic system.- 8.8 Further remarks, limits of applicability.- 9 Concluding Remarks and Suggestions for Future Research.- A Discretization and Stencils.