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Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations

verfasst von: Willem Hundsdorfer, Jan Verwer

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Professional Book Archive

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This book deals with numerical methods for solving partial differential equa tions (PDEs) coupling advection, diffusion and reaction terms, with a focus on time-dependency. A combined treatment is presented of methods for hy perbolic problems, thereby emphasizing the one-way wave equation, meth ods for parabolic problems and methods for stiff and non-stiff ordinary dif ferential equations (ODEs). With regard to time-dependency we have at tempted to present the algorithms and the discussion of their properties for the three different types of differential equations in a unified way by using semi-discretizations, i. e. , the method of lines, whereby the PDE is trans formed into an ODE by a suitable spatial discretization. In addition, for hy perbolic problems we also discuss discretizations that use information based on characteristics. Due to this combination of methods, this book differs substantially from more specialized textbooks that deal exclusively with nu merical methods for either PDEs or ODEs. We treat integration methods suitable for both classes of problems. This combined treatment offers a clear advantage. On the one hand, in the field of numerical ODEs highly valuable methods and results exist which are of practical use for solving time-dependent PDEs, something which is often not fully exploited by numerical PDE researchers. Although many problems can be solved by Euler's method or the Crank-Nicolson method, better alter natives are often available which can significantly reduce the computational effort needed to solve practical problems.

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Inhaltsverzeichnis

Frontmatter

I. Basic Concepts and Discretizations

Abstract

This chapter gives a first introduction to the numerical solution of time-dependent advection-diffusion-reaction problems. Our goal in this chapter is to discuss important basic concepts and discretizations for ordinary differential equations and for advection and diffusion equations in one spatial dimension. More advanced material will be treated in later chapters.

Willem Hundsdorfer, Jan Verwer

II. Time Integration Methods

Abstract

For the numerical solution of initial value problems for systems of ODEs there are many methods available, such as Runge-Kutta methods and linear multistep methods. In this chapter we give examples of methods which are of interest in the discretization of time-dependent PDEs. We will confine ourselves to methods having a low to moderate order. Further we pay attention to properties of specific interest to PDEs, namely the positivity property and the accuracy behaviour of Runge-Kutta methods for initial-boundary value problems. Excellent general references on ODE methods are Lambert (1991), Hairer, N0rsett & Wanner (1993) and Hairer & Wanner (1996).

Willem Hundsdorfer, Jan Verwer

III. Advection-Diffusion Discretizations

Abstract

In this third chapter we return to the discretization of advection-diffusion problems. We will treat here a number of special subjects that are supplementary to the more introductory material of Chapter I.

Willem Hundsdorfer, Jan Verwer

IV. Splitting Methods

Abstract

For many PDE problems in higher space dimension, such as the advection-diffusion-reaction system

$$ {u_t} + \nabla \cdot \left( {\underline a u} \right) = \nabla \cdot \left( {D\nabla u} \right) + f\left( u \right),$$

it is in general inefficient or infeasible to apply one and the same integration formula to the different parts of the system. For example, the chemistry can be very stiff, which calls for an implicit ODE method. On the other hand, if the advection is discretized in space using a limiter, then explicit methods are often much more suitable for that part of the equation. Moreover, use of a single implicit integration formula for the whole problem readily leads to a nonlinear algebraic system too large to handle due to the simultaneous coupling over the species and over space. In such cases a more tuned approach based on an appropriate form of splitting is advocated. The general idea behind splitting is breaking down a complicated problem into smaller parts for the sake of time stepping, such that the different parts can be solved efficiently with suitable integration formulas.

Willem Hundsdorfer, Jan Verwer

V. Stabilized Explicit Runge-Kutta Methods

Abstract

In this chapter we discuss special purpose explicit Runge-Kutta methods for systems of ODEs in ℝ^m

$$ w'\left( t \right) = F\left( {t,w\left( t \right)} \right),\quad t >0,\quad w\left( 0 \right) = {w_0},$$

representing semi-discrete, multi-space dimensional parabolic problems. Often, parabolic problems give rise to stiff systems having a symmetric Jacobian matrix ∂F(t,w)/∂wwith a spectral radius proportional to h ^-2, hrepresenting a spatial mesh width. Standard explicit methods are then highly inefficient due to their severe stability constraint, see Section II.1.4. On the other hand, unconditionally stable implicit methods, like backward Euler or the implicit trapezoidal rule, do require one or more linear or nonlinear algebraic system solutions at each integration step, which can become costly in higher space dimension.

Willem Hundsdorfer, Jan Verwer

Backmatter

Titel: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations
verfasst von: Willem Hundsdorfer
Jan Verwer
Copyright-Jahr: 2003
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-09017-6
Print ISBN: 978-3-642-05707-6
DOI: https://doi.org/10.1007/978-3-662-09017-6