##
**Robust computational techniques for boundary layers.**
*(English)*
Zbl 0964.65083

Applied Mathematics and Mathematical Computation. 16. Boca Raton, FL: Chapman & Hall/ CRC Press. xvi, 254 p. (2000).

The authors deal with the numerical solution of boundary-value problems for singularly perturbed differential equations whose exact solution possesses boundary layers. Linear problems are in the centre of the presentation, and so nonlinear differential equations are only considered marginally.

The book’s emphasis lies on giving evidence that a simple upwind-scheme on Shishkin-type mesh is the appropriate numerical method for resolving boundary layers. In this way, other numerical methods are more or less neglected.

The authors intend to reach not only mathematicians but also engineers and scientists. They prefer, therefore, a more heuristic style than a rigorous analytical presentation, and provide the reader with many computational examples including many tables and figures. Proofs are only carried out for linear, one-dimensional problems. This is not a drawback as it enables to focus on the main ideas and principles. Although the book is clearly written and easy to read, it, sometimes, is not clear who is the addressee: the novice in this field or those who are more acquainted with the topic.

The book contains more than 250 pages in 12 chapters. It starts with a discussion of which norm is suitable for error estimates (that tries to convince the reader of the global maximum norm as the only useful one). It follows the definition of a robust layer-resolving method that is a monotone (and thus non-oscillating) and with respect to the global (and not discrete) maximum norm parameter-uniformly convergent number method. In the sequel, the authors use two approaches for proving that a numerical scheme is robust layer-resolving: the analytical one, which is restricted to a few model problems, and the experimental one for concrete examples. In both, the convergence order and the error constant are determined and checked whether they are uniform with respect to the perturbation parameter.

In Chapter 2 (23 pp.), the discretization of linear convection-diffusion problems on a uniform mesh is analyzed (centred and simple upwind finite differences, fittered operator method). Unfortunately, the authors do not tell the reader that upwinding and fitting means centred difference for a modified equation which reflects the general idea of stabilization by adding artificial viscosity. The authors use again the concept of \(\varepsilon\)-uniform convergence only with respect to the global maximum norm. This is somewhat confusing at this stage as the numerical scheme only yields approximations at grid points. So the fitted Il’in-Allen-Southwell scheme is “in the global sense of this book” not \(\varepsilon\)-uniformly convergent although it is with respect to the discrete maximum norm. Only later in the text, it is shown that the scheme together with piecewise linear interpolation is not \(\varepsilon\)-uniformly convergent with respect to the global maximum norm.

Chapter 3 (36 pp.) deals with piecewise-uniform fitted meshes of Shishkin-type. The main objective is to prove that the simple upwind finite difference method on the Shishkin-type mesh together with piecewise linear interpolation is a robust layer-resolving method for the linear convection-diffusion problem in one dimension with Dirichlet as well as Neumann boundary conditions.

Disadvantages of the non-monotone centred difference scheme on piecewise-uniform meshes are discussed in Chapter 4 (19 pp).

It follows the study of two-dimensional convection-diffusion problems in a moving medium (Chapter 5, 28 pp), with frictionless walls (Chapter 6, 25 pp), and with no-slip boundary conditions (Chapter 7, 9 pp). The error constant and convergence order of a scheme are then numerically estimated (Chapter 8, 17 pp), and in Chapter 9 (16 pp), again non-monotone methods are discussed now for the two-dimensional case.

In Chapter 10 (18 pp), the authors consider semilinear reaction-diffusion problems (with quadratic or cubic nonlinearity).

The final two chapters (40 pp) deal with the Prandtl flow past a flat plate. The authors show how the Prandtl flow reduces to the Blasius equation which is, indeed, equivalent to a second-order singular perturbed ordinary differential equation. So the numerical methods discussed earlier can be applied. The results are then compared with a direct discretization.

The bibliography with about 60 entries is rather short but includes topical as well as original research references beside a series of some older and some quite interesting historical references. Neverthless, a pointer to the relevant literature is occasionally missing when reading the book.

The subject and name index with about 60 main entries on one and a half page gives sufficient support but a more comprehensive one would be of more use for the reader who is new in this field and who needs to find a path through the terminology.

At the very end, the book can be thoroughly suggested for those who are interested in, and not so familiar with, the numerical solution of singularly perturbed problems. It might be complementary to the more comprehensive presentation in the book of H.-G. Roos, M. Stynes and L. Tobiska [Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. Springer-Verlag (1996; Zbl 0844.65075)] and to the introductory monograph of K. W. Morton [Numerical solution of convection-diffusion problems. Chapman & Hall, London (1996; Zbl 0861.65070)].

The book’s emphasis lies on giving evidence that a simple upwind-scheme on Shishkin-type mesh is the appropriate numerical method for resolving boundary layers. In this way, other numerical methods are more or less neglected.

The authors intend to reach not only mathematicians but also engineers and scientists. They prefer, therefore, a more heuristic style than a rigorous analytical presentation, and provide the reader with many computational examples including many tables and figures. Proofs are only carried out for linear, one-dimensional problems. This is not a drawback as it enables to focus on the main ideas and principles. Although the book is clearly written and easy to read, it, sometimes, is not clear who is the addressee: the novice in this field or those who are more acquainted with the topic.

The book contains more than 250 pages in 12 chapters. It starts with a discussion of which norm is suitable for error estimates (that tries to convince the reader of the global maximum norm as the only useful one). It follows the definition of a robust layer-resolving method that is a monotone (and thus non-oscillating) and with respect to the global (and not discrete) maximum norm parameter-uniformly convergent number method. In the sequel, the authors use two approaches for proving that a numerical scheme is robust layer-resolving: the analytical one, which is restricted to a few model problems, and the experimental one for concrete examples. In both, the convergence order and the error constant are determined and checked whether they are uniform with respect to the perturbation parameter.

In Chapter 2 (23 pp.), the discretization of linear convection-diffusion problems on a uniform mesh is analyzed (centred and simple upwind finite differences, fittered operator method). Unfortunately, the authors do not tell the reader that upwinding and fitting means centred difference for a modified equation which reflects the general idea of stabilization by adding artificial viscosity. The authors use again the concept of \(\varepsilon\)-uniform convergence only with respect to the global maximum norm. This is somewhat confusing at this stage as the numerical scheme only yields approximations at grid points. So the fitted Il’in-Allen-Southwell scheme is “in the global sense of this book” not \(\varepsilon\)-uniformly convergent although it is with respect to the discrete maximum norm. Only later in the text, it is shown that the scheme together with piecewise linear interpolation is not \(\varepsilon\)-uniformly convergent with respect to the global maximum norm.

Chapter 3 (36 pp.) deals with piecewise-uniform fitted meshes of Shishkin-type. The main objective is to prove that the simple upwind finite difference method on the Shishkin-type mesh together with piecewise linear interpolation is a robust layer-resolving method for the linear convection-diffusion problem in one dimension with Dirichlet as well as Neumann boundary conditions.

Disadvantages of the non-monotone centred difference scheme on piecewise-uniform meshes are discussed in Chapter 4 (19 pp).

It follows the study of two-dimensional convection-diffusion problems in a moving medium (Chapter 5, 28 pp), with frictionless walls (Chapter 6, 25 pp), and with no-slip boundary conditions (Chapter 7, 9 pp). The error constant and convergence order of a scheme are then numerically estimated (Chapter 8, 17 pp), and in Chapter 9 (16 pp), again non-monotone methods are discussed now for the two-dimensional case.

In Chapter 10 (18 pp), the authors consider semilinear reaction-diffusion problems (with quadratic or cubic nonlinearity).

The final two chapters (40 pp) deal with the Prandtl flow past a flat plate. The authors show how the Prandtl flow reduces to the Blasius equation which is, indeed, equivalent to a second-order singular perturbed ordinary differential equation. So the numerical methods discussed earlier can be applied. The results are then compared with a direct discretization.

The bibliography with about 60 entries is rather short but includes topical as well as original research references beside a series of some older and some quite interesting historical references. Neverthless, a pointer to the relevant literature is occasionally missing when reading the book.

The subject and name index with about 60 main entries on one and a half page gives sufficient support but a more comprehensive one would be of more use for the reader who is new in this field and who needs to find a path through the terminology.

At the very end, the book can be thoroughly suggested for those who are interested in, and not so familiar with, the numerical solution of singularly perturbed problems. It might be complementary to the more comprehensive presentation in the book of H.-G. Roos, M. Stynes and L. Tobiska [Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. Springer-Verlag (1996; Zbl 0844.65075)] and to the introductory monograph of K. W. Morton [Numerical solution of convection-diffusion problems. Chapman & Hall, London (1996; Zbl 0861.65070)].

Reviewer: Etienne Emmrich (Berlin)

### MSC:

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

35K57 | Reaction-diffusion equations |

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

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

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

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

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

34E15 | Singular perturbations for ordinary differential equations |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35B25 | Singular perturbations in context of PDEs |

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

76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |

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

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