9 Conservation Laws 9.1 Introduction 9.2 Theory of Scalar Conservation Laws 9.2.1 Shock Formation 9.2.2 Weak Solutions 9.2.3 Discontinuous Solutions 9.2.4 The Entropy Condition 9.2.5 Solution of Scalar Conservation Laws 9.3 Theory of Systems of Conservation Laws 9.3.1 Solution of Riemann Problems 9.4 Computational Interlude VI 9.5 Numerical Solution of Conservation Laws 9.5.1 Introduction 9.6 Difference Schemes for Conservation Laws 9.6.1 Consistency 9.6.2 Conservative Schemes 9.6.3 Discrete Conservation 9.6.4 The Courant-Friedrichs-Levy Condition 9.6.5 Entropy 9.7 Difference Schemes for Scalar Conservation Laws 9.7.1 Definitions 9.7.2 Theorems 9.7.3 Godunov Scheme 9.7.4 High Resolution Schemes 9.7.5 Flux-Limiter Methods 9.7.6 Slope-Limiter Methods 9.7.7 Modified Flux Method 9.8 Difference Schemes for K-System Conservation Laws 9.9 Godunov Schemes 9.9.1 Godunov Scheme for Linear K-System Conservation Laws 9.9.2 Godunov Schemes for K-System Conservation Laws 9.9.3 Approximate Riemann Solvers: Theory 9.9.4 Approximate Riemann Solvers: Application 9.10 High Resolution Schemes for Linear K-System Conservation Laws 9.10.1 Flux-Limiter Schemes for Linear K-System Conservation Laws 9.10.2 Slope-Limiter Schemes for Linear K-System Conservation Laws 9.10.3 A Modified Flux Scheme for Linear K-System Conservation Laws 9.10.4 High Resolution Schemes for K-System Conservation Laws 9.11 Implicit Schemes 9.12 Difference Schemes for Two Dimensional Conservation Laws 9.12.1 Some Computational Examples 9.12.2 Some Two Dimensional High Resolution Schemes 9.12.3 The Zalesak-Smolarkiewicz Scheme 9.12.4 A Z-S Scheme for Nonlinear Conservation Laws 9.12.5 Two Dimensional K-System Conservation Laws10 Elliptic Equations 10.1 Introduction 10.2 Solvability of Elliptic Difference Equations: Dirichlet Boundary Conditions 10.3 Convergence of Elliptic Difference Schemes: Dirichlet Boundary Conditions 10.4 Solution Schemes for Elliptic Difference Equations: Introduction 10.5 Residual Correction Methods 10.5.1 Analysis of Residual Correction Schemes 10.5.2 Jacobi Relaxation Scheme 10.5.3 Analysis of the Jacobi Relaxation Scheme 10.5.4 Stopping Criteria 10.5.5 Implementation of the Jacobi Scheme 10.5.6 Gauss-Seidel Scheme 10.5.7 Analysis of the Gauss-Seidel Relaxation Scheme 10.5.8 Successive Overrelaxation Scheme 10.5.9 Elementary Analysis of SOR Scheme 10.5.10 More on the SOR Scheme 10.5.11 Line Jacobi, Gauss-Seidel and SOR Schemes 10.5.12 Approximating: Reality 10.6 Elliptic Difference Equations: Neumann Boundary Conditions 10.6.1 First Order Approximation 10.6.2 Second Order Approximation 10.6.3 Second Order Approximation on an Offset Grid 10.7 Numerical Solution of Neumann Problems 10.7.1 Introduction 10.7.2 Residual Correction Schemes 10.7.3 Jacobi and Gauss-Seidel Iteration 10.7.4 SOR Scheme 10.7.5 Approximation of
10.7.6 Implementation: Neumann Problems 10.8 Elliptic Difference Equations: Mixed Problems 10.8.1 Introduction 10.8.2 Mixed Problems: Solvability 10.8.3 Mixed Problems: Implementation 10.9 Elliptic Difference Equations: Polar Coordinates 10.10 Multigrid 10.10.1 Introduction 10.10.2 Smoothers 10.10.3 Grid Transfers 10.10.4 Multigrid Algorithm 10.11 Computational Interlude VII 10.11.1 Blocking Out: Irregular Regions 10.11.2 HW0.0.4 10.12 ADI Schemes 10.13 Conjugate Gradient Scheme 10.13.1 Preconditioned Conjugate Gradient 10.13.2 SSOR as a Preconditioner 10.13.2 Implementation 10.14 Using Iterative Methods to Solve Time Dependent Problems 10.15 Using FFT's to Solve Elliptic Problems 10.16 Computational Interlude VIII
11 Irregular Regions and Grids 11.1 Introduction 11.2 Irregular Geometries 11.2.1 Blocking Out 11.2.2 Map the Region 11.2.3 Grid Generation 11.3 Grid Refinement 11.3.1 Grid Refinement: Explicit Schemes for Hyperbolic Problems 11.3.2 Grid Refinement for Implicit Schemes 11.4 Unstructured Grids
James Thomas
10/18/1999