High order difference methods for time dependent PDE.

*(English)*Zbl 1146.65064
Springer Series in Computational Mathematics 38. Berlin: Springer (ISBN 978-3-540-74992-9/hbk). xv, 334 p. (2008).

This book presents the theory and construction principles of high order finite difference methods (FDM) for numerical solving of time dependent partial differential equations. Special attention is paid both to general techniques and appropriate examples, illustrating the advantages of the methods.

The author begins with simple model problems on wave propagation, parabolic and Schrödinger type equations to show the extent of effectiveness of high order FDM. Next, a survey of the three basic techniques: Fourier analysis, energy method and Laplace transform is exposed. These techniques are used in subsequent study of well-posedness and stability of various problems and methods. Order of accuracy, convergence rate, approximations in space, time, coupled space-time and boundary conditions are also considered in details. A separate chapter discusses wave propagation applications, where the benefit of high order methods is more evident. Many types of finite difference schemes are completely studied and numerical experiments and graphs are presented. The next chapter is concentrated on the application of a recently developed fourth order FDM for solving incompressible flow and Navier-Stokes equations. Some basic ideas and illustrations are also given to the construction of high order methods for nonlinear shock problems with smooth solutions. In the last chapter a brief introduction in other numerical methods, such as finite element method, spectral method etc. is presented.

The book is written in a clear and comprehensive manner. It is recommended to researchers, PD students and readers interested in effective methods for numerical solving of partial differential equations.

The author begins with simple model problems on wave propagation, parabolic and Schrödinger type equations to show the extent of effectiveness of high order FDM. Next, a survey of the three basic techniques: Fourier analysis, energy method and Laplace transform is exposed. These techniques are used in subsequent study of well-posedness and stability of various problems and methods. Order of accuracy, convergence rate, approximations in space, time, coupled space-time and boundary conditions are also considered in details. A separate chapter discusses wave propagation applications, where the benefit of high order methods is more evident. Many types of finite difference schemes are completely studied and numerical experiments and graphs are presented. The next chapter is concentrated on the application of a recently developed fourth order FDM for solving incompressible flow and Navier-Stokes equations. Some basic ideas and illustrations are also given to the construction of high order methods for nonlinear shock problems with smooth solutions. In the last chapter a brief introduction in other numerical methods, such as finite element method, spectral method etc. is presented.

The book is written in a clear and comprehensive manner. It is recommended to researchers, PD students and readers interested in effective methods for numerical solving of partial differential equations.

Reviewer: Snezhana Gocheva-Ilieva (Plovdiv)

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

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

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

35Q30 | Navier-Stokes equations |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M20 | Finite difference methods applied to problems in fluid mechanics |

44A10 | Laplace transform |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |