1st Edition
Conservative Finite-Difference Methods on General Grids
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.
Governing Equations
Elliptic Equations
Heat Equation
Equation of Gas Dynamic in Lagrangian Form
The Main Ideas of Finite-Difference Algorithms
1-D Case
2-D Case
Methods of Solution of Systems of Linear Algebraic Equation
Methods of Solution of Systems of Nonlinear Equations
METHOD OF SUPPORT-OPERATORS
Main Stages
The Elliptic Equations
Gas Dynamic Equations
System of Consistent Difference Operators in 1-D
Inner Product in Spaces of Difference Functions and Properties of Difference Operators
System of Consistent Difference Operators in 2-D
THE ELLIPTIC EQUATIONS
Introduction
Continuum Elliptic Problems with Dirichlet Boundary Conditions
Continuum Elliptic Problems with Robin Boundary Conditions
One-Dimensional Support Operator Algorithms
Nodal Discretization of Scalar Functions and Cell-Centered Discretization of Vector Functions
Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions
Numerical Solution of Test Problems
Two-Dimensional Support Operator Algorithms
Nodal Discretization of Scalar Functions and Cell-Valued Discretization of Vector Functions
Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions
Numerical Solution of Test Problems
Conclusion
Two-Dimensional Support Operator Algorithms
Discretization
Spaces of Discrete Functions
The Prime Operator
The Derived Operator
Multiplication by a Matrix and the Operator D
The Difference Scheme for the Elliptic Operator
The Matrix Problem
Approximation and Convergence Properties
HEAT EQUATION
Introduction
Finite-Difference Schemes for Heat Equation in 1-D
Finite-Difference Schemes for Heat Equation in 2-D
LAGRANGIAN GAS DYNAMICS
Kinematics of Fluid Motions
Integral Form of Gas Dynamics Equations
Integral Equations for One Dimensional Case
Differential Equations of Gas Dynamics in Lagrangian Form
The Differential Equations in 1D. Lagrange Mass Variables
The Statements of Gas Dynamics Problems in Lagrange Variables
Different Forms of Energy Equation
Acoustic Equations
Reference Information
Characteristic Form of Gas Dynamics Equations
Riemann's Invariants
Discontinuous Solutions
Conservation Laws and Properties of First Order Invariant Operators
Finite-Difference Algorithm in 1D
Discretization in 1D
Discrete Operators in 1D
Semi-Discrete Finite-Difference Scheme in 1D
Fully Discrete, Explicit, Computational Algorithm
Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme
Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme
Stability Conditions
Homogeneous Finite-Difference Schemes. Artificial Viscosity
Artificial Viscosity in 1D
Numerical Example
Finite Difference Algorithm in 2D
Discretization in 2D
Discrete Operators in 2D
Semi-Discrete Finite-Difference Scheme in 2D
Stability Conditions
Finite-Difference Algorithm in 2D
Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme
Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme
Artificial Viscosity in 2D
Numerical Example
APPENDIX: FORTRAN CODE DIRECTORY
General Description of Structure of Directories on the Disk
Programs for Elliptic Equations
Programs for 1D Equations
Programs for 2D Equations
Programs for Heat Equations
Programs for 1D Equations
Programs for 2D Equations
Programs for Gas Dynamics Equations
Programs for 1D Equations
Programs for 2D Equations
Bibliography
Biography
Shashkov, Mikhail
