Skip to main content
main-content

Über dieses Buch

The stimulus for the present work is the growing need for more accurate numerical methods. The rapid advances in computer technology have not provided the resources for computations which make use of methods with low accuracy. The computational speed of computers is continually increasing, while memory still remains a problem when one handles large arrays. More accurate numerical methods allow us to reduce the overall computation time by of magnitude. several orders The problem of finding the most efficient methods for the numerical solution of equations, under the assumption of fixed array size, is therefore of paramount importance. Advances in the applied sciences, such as aerodynamics, hydrodynamics, particle transport, and scattering, have increased the demands placed on numerical mathematics. New mathematical models, describing various physical phenomena in greater detail than ever before, create new demands on applied mathematics, and have acted as a major impetus to the development of computer science. For example, when investigating the stability of a fluid flowing around an object one needs to solve the low viscosity form of certain hydrodynamic equations describing the fluid flow. The usual numerical methods for doing so require the introduction of a "computational viscosity," which usually exceeds the physical value; the results obtained thus present a distorted picture of the phenomena under study. A similar situation arises in the study of behavior of the oceans, assuming weak turbulence. Many additional examples of this type can be given.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
The widespread use of comupters in problem solving, in both science and technology, has stimulated the development of many numerical algorithms. Most of these algorithms are based on the reduction of an initial differential probelm, to the same problems in linear algebra. At present this method is the most widely used means for solving applied problems. It is natural that the dimensionality of the linear algebra problems obtained depends on the discretization parameter, the latter being generally the finite difference mesh-size. Thus, more accurate solutions require smaller discretization parameters. By decreasing the parameter we generally increase the number of linear equations, which usually results in an increase in the amount of computation time.
G. I. Marchuk, V. V. Shaidurov

Chapter 1. General Properties

Abstract
We begin this chapter with a concrete example. In this example we use the solutions of a difference scheme of rather low accuracy and achieve a more accurate approximate solution using the Richardson extrapolation with higher-order differences. We then extend these results to abstract problems: we first, formulate a sufficient condition for the existence of an expansion of the approximate solution in powers of the difference net mesh-size; then we prove that this expansion allows us to use either the Richardson extra­polation or higher-order difference corrections.
G. I. Marchuk, V. V. Shaidurov

Chapter 2. First-Order Ordinary Differential Equations

Abstract
When one solves differential equations numerically one faces the problem of finding those solutions with as great an accuracy as possible in terms of the mesh-size of the net on which the reduction of the differential equation to a difference equation is carried out. At present many methods exist for finding approximate solutions with a given degree of accuracy when the solution of the differential equation is smooth. The Runge-Kutta method is one of the most widely used since it allows one to create an algorithm for the solution of the problem in a simple and straightforward manner. The theoretical foundations for the construction of the algorithm are well understood as well, making this a most attractive method of numerical mathematics.
G. I. Marchuk, V. V. Shaidurov

Chapter 3. The One-Dimensional Stationary Diffusion Equation

Abstract
The diffusion equation is of great interest because of its application to various fields of science. This equation is of special interest in the theory of nuclear reactors, where the diffusion approximation of the transport equation is of great importance. In ecology the computation of the diffusion of industrial aerosols is of great importance. Important diffusion problems occur in physics, chemistry, geophysics, and other fields of science. Because of this we shall focus on various formulations of problems related to the diffusion equation.
G. I. Marchuk, V. V. Shaidurov

Chapter 4. Elliptic Equations

Abstract
Boundary-value problems for elliptic equations are the most common problems of mathematical physics. They have many practical applications. They also appear in the reduced forms of parabolic and hyperbolic equations. Therefore it is quite natural to pay particular attention to problems related to elliptic operators. Here we will not focus on well-known facts connected with the statement of the boundary-value problem of elliptic type, and the dependence of the solutions on the properties of the input data. This information can be found in the literature. We will give only the necessary facts from the theory, and focus on the application of the various approaches to improving the difference solutions to such problems. In addition to linear problems a solvable nonlinear problem will also be considered. This nonlinear problem, as well as a simple diffraction problem, have been chosen because we wanted to demonstrate the improvement in the accuracy of the solutions for relatively simple problems. Our aim was also to acquaint the reader with basic algorithmic techniques used.
G. I. Marchuk, V. V. Shaidurov

Chapter 5. Nonstationary Problems

Abstract
Nonstationary equations tend to be more complicated than stationary ones. They have many applications in various fields of science and technology. When constructing numerical algorithms one must deal with spatial variables as well as with the time variable. The question of finding a corrector for an approximate solution on a sequence of nets is considerably more complicated. Nevertheless in a great number of cases such correctors can be found. These questions will now occupy us. In this book we will not treat high order schemes which use another method, nor will we discuss other algorithms even though they also give successful results. The reader is referred to [4, 131, 51, 87, 112, 114, 42, 146], for further discussion.
G. I. Marchuk, V. V. Shaidurov

Chapter 6. Extrapolation for Algebraic Problems and Integral Equations

Abstract
The method of extrapolation can be applied to problems other than the numerical solution of differential equations. This method, with some modifications, can also be applied to other problems of numerical mathematics.
G. I. Marchuk, V. V. Shaidurov

Chapter 7. Appendix

Abstract
In this chapter we present some simple results often used in the previous chapters.
G. I. Marchuk, V. V. Shaidurov

Backmatter

Weitere Informationen