Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations
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This book describes numerical methods for partial differential equations (PDEs) coupling advection, diffusion and reaction terms, encompassing methods for hyperbolic, parabolic and stiff and nonstiff ordinary differential equations (ODEs). The emphasis lies on time-dependent transport-chemistry problems, describing e.g. the evolution of concentrations in environmental and biological applications. Along with the common topics of stability and convergence, much attention is paid on how to prevent spurious, negative concentrations and oscillations, both in space and time. Many of the theoretical aspects are illustrated by numerical experiments on models from biology, chemistry and physics. A unified approach is followed by emphasizing the method of lines or semi-discretization. In this regard this book differs substantially from more specialized textbooks which deal exclusively with either PDEs or ODEs. This book treats integration methods suitable for both classes of problems and thus is of interest to PDE researchers unfamiliar with advanced numerical ODE methods, as well as to ODE researchers unaware of the vast amount of interesting results on numerical PDEs. The first chapter provides a self-contained introduction to the field and can be used for an undergraduate course on the numerical solution of PDEs. The remaining four chapters are more specialized and of interest to researchers, practitioners and graduate students from numerical mathematics, scientific computing, computational physics and other computational sciences.
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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