Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems.

*(English)*Zbl 0844.65075
Springer Series in Computational Mathematics. 24. Berlin: Springer-Verlag. xvi, 348 p. (1996).

The development of robust numerical methods for singular perturbation problems is at a very early stage, and that fact is reflected in the content of this monograph. The book deals with convection-diffusion equations of four types, and the organization is according to increasing difficulty. Specifically, the topics are 2-point boundary value problems for ordinary differential equations, parabolic problems in one spatial dimension, elliptic boundary value problems, and incompressible Navier-Stokes problems. For each of these topics the discussion starts with a discussion of the analytic behavior of the solutions. This is followed by a survey of numerical methods, along with as much analysis of stability and accuracy as can reasonably be given.

The state of the field is that there exist numerical methods which are very effective for some limited class of problems. There also exist more general methods (such as artificial viscosity), but their accuracy can be disappointing. The book gives thorough discussions of both types of methods. The only negative comment from the reviewer is that the authors could have given more emphasis to advice to the novice about when to use which method. Generally, one uses a regular grid and a numerical method with ample viscosity when one knows little about the behavior of the solution of the problem, and one builds in adapted grids and special basis functions as one learns more about the location and behavior of internal and boundary layers. It would be desirable to have software which carries out such a process adaptively, but such software now exists only for very limited classes of problems.

The state of the field is that there exist numerical methods which are very effective for some limited class of problems. There also exist more general methods (such as artificial viscosity), but their accuracy can be disappointing. The book gives thorough discussions of both types of methods. The only negative comment from the reviewer is that the authors could have given more emphasis to advice to the novice about when to use which method. Generally, one uses a regular grid and a numerical method with ample viscosity when one knows little about the behavior of the solution of the problem, and one builds in adapted grids and special basis functions as one learns more about the location and behavior of internal and boundary layers. It would be desirable to have software which carries out such a process adaptively, but such software now exists only for very limited classes of problems.

Reviewer: G.Hedstrom (Livermore)

##### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

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

65L12 | Finite difference and finite volume methods for ordinary differential equations |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

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

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

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

34E15 | Singular perturbations, general theory for ordinary differential equations |

35B25 | Singular perturbations in context of PDEs |

34B05 | Linear boundary value problems for ordinary differential equations |