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**Numerical solution of time-dependent advection-diffusion-reaction equations.**
*(English)*
Zbl 1030.65100

Springer Series in Computational Mathematics. 33. Berlin: Springer. x, 471 p. (2003).

Time dependent partial differential equations (PDEs) that couple advection, diffusion and reaction terms are investigated in this book especially from numerical point of view. With regard to time dependency algorithms and discussion of their properties are presented for different types of differential equations. For example the method of lines where the partial differential equation is transformed into a system of ordinary differential equations (ODEs) by a suitable spatial discretization. For hyperbolic problems discretizations that use information based on characteristics are also discussed. So there is a combination of methods for both PDEs and ODEs in this book. This approach has a clear advantage because in the field of numerical ODEs highly valuable methods and results exist which are of practical use for solving time dependent PDEs and on the other hand vast amount of highly interesting results on discretization methods for PDEs can be usefull also for ODEs researchers.

From the application point of view, the authors have focused mainly to so-called transport-chemistry problems: i.e problems where the transport part is based on advection and diffusion processes and the chemistry part on chemical reaction processes modelled by ordinary differential equations. These modells are often used in environmental modelling in connection with pollution of atmospheric air, surface water and groundwater. Similar problems arise also in mathematical bilology in study bacterial growth, tumor growth and related biochemical phenomena.

The book consists of five chapters. The first one presents an introduction to the wide field of numerical solution of evolutionary PDEs. It is devoted mainly to students of applied and numerical analysis or scientists less educated in numerical analysis. The other chapters are more precise and specialized on certain subjects, and they are very useful for more experienced readers. They are chosen for their practical relevance for solving advection-diffusion-reaction problems.

The second chapter is devoted to time integration methods. Examples which are of interest in the discretization of time-dependent PDE are presented, especially Runge-Kutta methods and linear multistep methods. Rosenbrock methods as a special case of Runge-Kutta methods are also presented. Stability and convergence conditions are proposed. Error analysis and positivity property is included too. At the end of this chapter several illustrated examples are involved.

In the third chapter the authors return back to the spatial discretization problem for advection-diffusion problems. Several types of high order discrete fluxes are presented together with numerical experiments. The total variation diminishing property that prevents localized under and overshoots is discussed. As a non-uniform grid and for multtidimensional space problems finite volume and finite element methods are presented.

The fourth chapter deals with so-called splitting methods. The general idea in splitting is in substituting one complicated problem to two or more simple problems. Basic techniques for constructing the splitting operator for first and second order linear or nonlinear ODEs and advection-diffusion-reaction problems are presented. Separate sections are devoted to describing the main properties of LOD methods, ADI methods and IMEX methods together with a lot of numerical examples which illustrate the efficiency of the presented methods.

In last the chapter special explicit Runge-Kutta methods are discussed. The stabilized Runge-Kutta methods are explicit and thus avoid the solution of an algebraic system and possess extended real stability intervals with length proportional to \(s^2\), where \(s\) is number of stages. Two families of these methods are disscussed: Runge-Kutta-Chebyshev methods and orthogonal-Runge-Kutta-Chebyshev methods.

From the application point of view, the authors have focused mainly to so-called transport-chemistry problems: i.e problems where the transport part is based on advection and diffusion processes and the chemistry part on chemical reaction processes modelled by ordinary differential equations. These modells are often used in environmental modelling in connection with pollution of atmospheric air, surface water and groundwater. Similar problems arise also in mathematical bilology in study bacterial growth, tumor growth and related biochemical phenomena.

The book consists of five chapters. The first one presents an introduction to the wide field of numerical solution of evolutionary PDEs. It is devoted mainly to students of applied and numerical analysis or scientists less educated in numerical analysis. The other chapters are more precise and specialized on certain subjects, and they are very useful for more experienced readers. They are chosen for their practical relevance for solving advection-diffusion-reaction problems.

The second chapter is devoted to time integration methods. Examples which are of interest in the discretization of time-dependent PDE are presented, especially Runge-Kutta methods and linear multistep methods. Rosenbrock methods as a special case of Runge-Kutta methods are also presented. Stability and convergence conditions are proposed. Error analysis and positivity property is included too. At the end of this chapter several illustrated examples are involved.

In the third chapter the authors return back to the spatial discretization problem for advection-diffusion problems. Several types of high order discrete fluxes are presented together with numerical experiments. The total variation diminishing property that prevents localized under and overshoots is discussed. As a non-uniform grid and for multtidimensional space problems finite volume and finite element methods are presented.

The fourth chapter deals with so-called splitting methods. The general idea in splitting is in substituting one complicated problem to two or more simple problems. Basic techniques for constructing the splitting operator for first and second order linear or nonlinear ODEs and advection-diffusion-reaction problems are presented. Separate sections are devoted to describing the main properties of LOD methods, ADI methods and IMEX methods together with a lot of numerical examples which illustrate the efficiency of the presented methods.

In last the chapter special explicit Runge-Kutta methods are discussed. The stabilized Runge-Kutta methods are explicit and thus avoid the solution of an algebraic system and possess extended real stability intervals with length proportional to \(s^2\), where \(s\) is number of stages. Two families of these methods are disscussed: Runge-Kutta-Chebyshev methods and orthogonal-Runge-Kutta-Chebyshev methods.

Reviewer: Angela Handlovičová (Bratislava)

### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

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

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

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

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

92E20 | Classical flows, reactions, etc. in chemistry |

80A32 | Chemically reacting flows |

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

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