Conservative Finite-Difference Methods on General Grids

Mikhail Shashkov

• Publisher: CRC Press
• Year: 2018
• Topic: Mathematics
• eBook ISBN:
  9781351458290
• Subtopic: Mathematics General
• Index: Mathematics

This new book deals with the construction of finite-difference (FD) algorithms for three main types of equations: elliptic equations, heat equations, and gas dynamic equations in Lagrangian form. These methods can be applied to domains of arbitrary shapes. The construction of FD algorithms for all types of equations is done on the basis of the support-operators method (SOM). This method constructs the FD analogs of main invariant differential operators of first order such as the divergence, the gradient, and the curl. This book is unique because it is the first book not in Russian to present the support-operators ideas. Conservative Finite-Difference Methods on General Grids is completely self-contained, presenting all the background material necessary for understanding. The book provides the tools needed by scientists and engineers to solve a wide range of practical engineering problems. An abundance of tables and graphs support and explain methods. The book details all algorithms needed for implementation. A 3.5" IBM compatible computer diskette with the main algorithms in FORTRAN accompanies text for easy use.

1 Introduction

The main goal of this book is to construct the finite-difference algorithms for solutions to differential equations of mathematical physics. To emphasize the relationship between these algorithms and the usual university course for equations of mathematical physics, we describe the typical equation and statements of boundary problems for it. We do not have room for a detailed description of physical problems which lead to the described equations in addition to the theory of boundary value problems so we refer readers to the following books related to these topics: [11], [26], [52], [133], [149].

1 Governing Equations

There are two main types of the physical process: non-stationary (time dependent) and stationary (which is not depend on time). The examples of non-stationary processes in this book are heat equations (equations of parabolic type) and equations of gas dynamics in Lagrangian form (the system of equations of hyperbolic type). The stationary processes are presented by the stationary heat equation (equations of elliptic type). Following is a more detailed description of these equations.

1.1 Elliptic Equations

The simplest example of equations of elliptic type is the Laplace equation

Δu=Σα=13∂2u∂xα2=0,⁢(1.1)

where u = u(x) is the unknown function of x, x = (x₁, x₂, x₃) is a point in 3-D Cartesian space.

The non-homogeneous equation

Δu=-f(x),

is called the Poisson equation. The problems associated with stationary heat transfer, the stationary diffusion process, and with electrostatics lead to the Laplace and Poisson equations.

Stationary distribution of temperature u = u(x₁, x₂, x₃) in homogeneous media can be described by the Poisson equation

Δu=-f/k,

where f(x) is the density of heat sources, and k = const > 0 is a coefficient of heat conductivity.

For non-homogeneous media the coefficient of heat conductivity depends on the position x, k = k(x) and instead of the Poisson equation we get

div(kgradu)=Σα=13∂∂xα(k(x)∂u∂xα)=-f(x),

where...

Table of contents

• Cover Page
• Half title
• Title Page
• Copyright
• Preface
• Acknowledgments
• About the Author
• Contents
• Notations
• 1 Introduction
• 2 Method of Support-Operators
• 3 The Elliptic Equations
• 4 The Heat Equation
• 5 Lagrangian Gas Dynamics
• 6 Conclusion
• 3 Programs for Heat Equations
• Index