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**Parallel and sequential methods for ordinary differential equations.**
*(English)*
Zbl 0838.65073

Oxford: Clarendon Press. xvi, 446 p. (1995).

This monograph is concerned with the very interesting topic on parallel methods for differential equations and steps into the breach. In recognition of the enormous amount of research now being devoted to scientific parallel computing, this monograph offers an important up-to date exposition of the current state of the art of numerical methods for solving differential equations in a parallel environment.

The book is in three parts. The first four chapters can be considered as introductory material but with new and stimulating subjects. In Chapters 5 and 6 the various different forms that parallelism can take when solving differential equations are discussed with the main focus being the analysis and construction of direct and iterative methods. The underlying concepts are parallelism across the method and parallelism across the steps. Chapters 7, 8 and 9 are devoted to a thorough description of waveform techniques for exploiting large scale parallelism, which means parallelism across the system.

Chapter 1 gives a discussion of parallel computing in general: the different types of architectures, case studies of various machines and an introduction to a number of parallel paradigms. Chapters 2 and 3 present a summary of existing sequential methods for solving ordinary differential equations. Because of the strong connection between differential equation algorithms and linear algebra techniques, Chapter 4 offers material on parallel methods for the solution of linear systems: direct and iterative methods, multigrid techniques, deflation techniques and adaptive preconditioning techniques. Chapters 5 and 6 are devoted, respectively, to direct methods and methods which exploit parallelism across the steps. Some implementational work of these algorithms is also presented. Chapter 7 gives a discussion and analysis of dynamic iteration techniques for differential equations, while Chapter 8 will examine what happens when numerical methods are used in conjugation with waveform relaxation algorithms. Chapter 9 is devoted to parallel implementational issues of waveform relaxation. A code based on block Jacobi waveform relaxation and VODE (a BDF/Adams code) using message-passing paradigms is described and numerical results are given. At the end a number of conclusions and thoughts on future directions are added.

The book is in three parts. The first four chapters can be considered as introductory material but with new and stimulating subjects. In Chapters 5 and 6 the various different forms that parallelism can take when solving differential equations are discussed with the main focus being the analysis and construction of direct and iterative methods. The underlying concepts are parallelism across the method and parallelism across the steps. Chapters 7, 8 and 9 are devoted to a thorough description of waveform techniques for exploiting large scale parallelism, which means parallelism across the system.

Chapter 1 gives a discussion of parallel computing in general: the different types of architectures, case studies of various machines and an introduction to a number of parallel paradigms. Chapters 2 and 3 present a summary of existing sequential methods for solving ordinary differential equations. Because of the strong connection between differential equation algorithms and linear algebra techniques, Chapter 4 offers material on parallel methods for the solution of linear systems: direct and iterative methods, multigrid techniques, deflation techniques and adaptive preconditioning techniques. Chapters 5 and 6 are devoted, respectively, to direct methods and methods which exploit parallelism across the steps. Some implementational work of these algorithms is also presented. Chapter 7 gives a discussion and analysis of dynamic iteration techniques for differential equations, while Chapter 8 will examine what happens when numerical methods are used in conjugation with waveform relaxation algorithms. Chapter 9 is devoted to parallel implementational issues of waveform relaxation. A code based on block Jacobi waveform relaxation and VODE (a BDF/Adams code) using message-passing paradigms is described and numerical results are given. At the end a number of conclusions and thoughts on future directions are added.

Reviewer: R.Scherer (Karlsruhe)

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65Y05 | Parallel numerical computation |

34A34 | Nonlinear ordinary differential equations and systems |