In computational mechanics, the first and quite often the most difficult part of a problem is the correct formulation of the problem. This is usually done in terms of differential equations. Once this formulation is accomplished, the translation of the governing differential equations into accurate, stable, and physically realistic difference equations can be a formidable task. By comparison, the numerical evaluation of these difference equations in order to obtain a solution is usually much simpler. The present notes are primarily concerned with the second task, that of deriving accurate, stable, and physically realistic difference equations from the governing differential equations. Procedures for the numerical evaluation of these difference equations are also presented. In later applications, the physical formulation of the problem and the properties of the numerical solution, especially as they are related to the numerical approximations inherent in the solution, are discussed. There are numerous ways to form difference equations from differential equations.

Klappentext

The primary purpose of this text is to illustrate the derivation of difference equations from differential equations using the volume integral method. In this method, the region of interest is divided into finite volume elements. The governing equations are then integrated over each of these elements. In order to form a difference equation, the terms in the resulting equations must be analytically approximated as well as possible. Various approaches to do this, including Taylor series expansions, have been developed and are illustrated in the text. The notes illustrate the applications of these ideas to ordinary and partial differential equations as well as some more specific applied problems. They have been used for an introductory course on numerical methods at the graduate level at the University of California, Santa Barbara.

Inhalt

1. Ordinary Differential Equations.- 1.1 Difference Equations by Means of Taylor Series.- 1.2 The Volume Integral Method.- 1.3 A More Interesting Example: The Convection-Diffusion Equation.- 1.4 Non-Uniformly Spaced Points.- 1.5 Boundary-Value Problems.- 1.6 Initial-Value Problems.- 1.7 A Higher-Order Boundary-Value Problem.- 2. Parabolic Equations.- 2.1 Standard Approximations for the Heat Equation.- 2.2 Stable, Explicit Approximations.- 2.3 Implicit Algorithms.- 2.4 Algorithms for Two-Dimensional Problems.- 2.5 Non-Uniformly Spaced Points.- 2.6 Polar Coordinates.- 3. Hyperbolic Equations.- 3.1 A Transport Equation.- 3.2 Other Linear, One-Dimensional, Time-Dependent Equations.- 3.3 Extensions to Two Space Dimensions.- 3.4 More on Open Boundary Conditions.- 3.5 Nesting and Wave Reflections for Non-Uniformly Spaced Points.- 3.6 Low-Speed, Almost Incompressible Flows.- 4. Elliptic Equations.- 4.1 Basic Difference Equations.- 4.2 Iterative Solutions.- 4.3 Singular Points.- 5. Applications.- 5.1 Currents in Aquatic Systems.- 5.2 The Transport of Fine-Grained Sediments in Aquatic Systems.- 5.3 Chemical Vapor Deposition.- 5.4 Free-Surface Flows Around Submerged or Floating Bodies.- General References.- Appendix A. Useful Taylor Series Formulas.- Appendix B. Solution of a System of Linear Algebraic Equations by Gaussian Elimination.