Parallel Multigrid Waveform Relaxation for Parabolic Problems [Paperback]

1 Introduction.- 1.1 Numerical simulation and parallel processing.- 1.2 The simulation of time-dependent processes.- 1.2.1 The serial bottleneck of time-marching.- 1.2.2 Accelerating the time-marching process.- 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.- 2.4.1 The contraction mapping principle.- 2.4.2 Waveform relaxation as a contraction map.- 2.4.3 On the order of accuracy.- 2.5 Convergence analysis for linear systems.- 2.5.1 Functional analysis preliminaries.- 2.5.2 Waveform relaxation for linear systems.- 2.5.3 Continuous-time convergence results.- 2.5.4 Discrete-time convergence results.- 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.- 3.2.1 A model problem: the two-dimensional heat equation.- 3.2.2 Standard waveform relaxation methods.- 3.2.3 Some convergence results.- 3.2.4 Numerical experiments.- 3.2.5 Further remarks.- 3.3 Linear multigrid acceleration.- 3.3.1 The multigrid principle.- 3.3.2 Linear multigrid waveform relaxation.- 3.4 Convergence analysis.- 3.4.1 Continuous-time convergence analysis.- 3.4.2 Discrete-time 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.- 4.4.1 The linear model problem.- 4.4.2 The time-periodic waveform iteration.- 4.4.3 The time-periodic integral operator.- 4.4.4 Convergence of the time-periodic waveform relaxation.- 4.4.5 The existence of convergent and divergent splittings.- 4.5 Analysis of the discrete-timelcN