Richardson Extrapolation

Practical Aspects and Applications

Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi

• Publisher: De Gruyter
• Year: 2017
• Print ISBN: 9783110516494
• eBook ISBN: 9783110531985
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Scientists and engineers are mainly using Richardson extrapolation as a computational tool for increasing the accuracy of various numerical algorithms for the treatment of systems of ordinary and partial differential equations and for improving the computational efficiency of the solution process by the automatic variation of the time-stepsizes. A third issue, the stability of the computations, is very often the most important one and, therefore, it is the major topic studied in all chapters of this book.
Clear explanations and many examples make this text an easy-to-follow handbook for applied mathematicians, physicists and engineers working with scientific models based on differential equations.

Contents
The basic properties of Richardson extrapolation
Richardson extrapolation for explicit Runge-Kutta methods
Linear multistep and predictor-corrector methods
Richardson extrapolation for some implicit methods
Richardson extrapolation for splitting techniques
Richardson extrapolation for advection problems
Richardson extrapolation for some other problems
General conclusions

1The basic properties of Richardson extrapolation

The basic principles on which the implementation of Richardson extrapolation in the numerical treatment of the important class of initial value problems for systems of ordinary differential equations (ODEs) is based are discussed in this chapter. It must immediately be stressed that this powerful approach can also be applied in the solution of systems of partial differential equations (PDEs). In the latter case, the device is often applied after the semi-discretization of the systems of PDEs. When the spatial derivatives are discretized, by applying, for example, finite differences or finite elements, the system of PDEs is transformed into a system of ODEs, which is as a rule very large. The application of Richardson extrapolation is then very straightforward, since it is based on the same rules as those used in the implementation of Richardson extrapolation for systems of ODEs, and this simple way of implementing this devise is very often used but its success is based on an assumption that the selected discretization of spatial derivatives is sufficiently accurate and, therefore, that errors resulting from spatial discretization are very small and will not interfere with errors resulting from the application of Richardson extrapolation in the solution of the semi-discretized problem. If this assumption is satisfied, then the results will in general be good, but problems will surely arise when the assumption is not satisfied. Discretization errors caused by the treatment of spatial derivatives must also be taken into account and the strict implementation of Richardson extrapolation for systems of PDEs will become considerably more complicated than that for systems of ODEs. Therefore, the direct application of Richardson extrapolation in the computer treatment of systems of PDEs deserves some special treatment. This is why the first five chapters of this book study only the application of Richardson extrapolation in the case where systems of ODEs are handled, while a description of the direct use of Richardson extrapolation for systems of PDEs is postponed and will be presented in Chapter 6.

The contents of the first chapter can be outlined as follows:

The initial value problem for systems of ODEs is introduced in Section 1.1. It is explained there when the solution of this problem exists and is unique. The assumptions that must be made in order to ensure existence and uniqueness of the solution are in fact not very restrictive, but it is stressed that some additional assumptions must be imposed when accurate numerical results are needed and, therefore, numerical methods for treatment of the initial value problems for systems of ODEs that have high orders of accuracy are to be selected and used. It must also be emphasized here that in Section 1.1 we are sketching only the main ideas. No details about the assumptions that must be made in order to ensure the existence and uniqueness of the solution of the initial value problems for systems of ODEs are needed, because this topic is not directly connected to the application of Richardson extrapolation in conjunction with different numerical methods for solving such problems. However, we provide references to several text books where such details are presented and discussed.

Some basic concepts that are related to the application of an arbitrary numerical method for solving initial value problems for systems of ODEs are briefly described in Section 1.2. It is explained there that computations are as a rule carried out step by step at the grid points of some set of values of the independent variable (which is the time-variable in many engineering...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Contents
• 1 The basic properties of Richardson extrapolation
• 2 Richardson extrapolation for explicit Runge–Kutta methods
• 3 Linear multistep and predictor-corrector methods
• 4 Richardson extrapolation for some implicit methods
• 5 Richardson extrapolation for splitting techniques
• 6 Richardson extrapolation for advection problems
• 7 Richardson extrapolation for some other problems
• 8 General conclusions
• References
• List of abbreviations
• Author index
• Subject index