Conservative Finite-Difference Methods on General Grids

Abstract

INTRODUCTION 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