NUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Transcription

1 NUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS JOHN A. TRANGENSTEIN Department of Mathematics, Duke University Durham, NC Ш CAMBRIDGE ЩР UNIVERSITY PRESS

2 Contents 1 Introduction to Partial Differential Equations 2 Scalar Hyperbolic Conservation Laws 2.1 Linear Advection Conservation Law on an Unbounded Domain Integral Form of the Conservation Law Advection-Diffusion Equation Advection Equation on a Half-Line Advection Equation on a Finite Interval 2.2 Linear Finite Difference Methods Basics of Discretization Explicit Upwind Differences Programs for Explicit Upwind Differences First Upwind Difference Program Second Upwind Difference Program Third Upwind Difference Program Fourth Upwind Difference Program Fifth Upwind Difference Program Explicit Downwind Differences Implicit Downwind Differences Implicit Upwind Differences Explicit Centered Differences 2.3 Modified Equation Analysis Modified Equation Analysis for Explicit Upwind Differences

3 Vlll Contents Modified Equation Analysis for Explicit Downwind Differences Modified Equation Analysis for Explicit Centered Differences Modified Equation Analysis Literature Consistency, Stability and Convergence Fourier Analysis of Finite Difference Schemes Constant Coefficient Equations and Waves Dimensionless Groups Linear Finite Differences and Advection Fourier Analysis of Individual Schemes L 2 Stability for Linear Schemes Lax Equivalence Theorem 55 о Measuring Accuracy and Efficiency 69 Nonlinear Scalar Laws Nonlinear Hyperbolic Conservation Laws Nonlinear Equations on Unbounded Domains Characteristics Development of Singularities Propagation of Discontinuities Traveling Wave Profiles Entropy Functions Oleinik Chord Condition Riemann Problems Galilean Coordinate Transformations Case Studies Traffic Flow Miscible Displacement Model Buckley-Leverett Model First-Order Finite Difference Methods Explicit Upwind Differences Lax-Friedrichs Scheme Timestep Selection Rusanov's Scheme Godunov's Scheme Comparison of Lax-Friedrichs, Godunov and Rusanov Nonreflecting Boundary Conditions Lax-Wendroff Process Other Second Order Schemes 132

4 Nonlinear Hyperbolic Systems 4.1 Theory of Hyperbolic Systems Contents Hyperbolicity and Characteristics Linear Systems Frames of Reference Useful Identities Change of Frame of Reference for Conservation Laws Change of Frame of Reference for Propagating Discontinuities Rankine-Hugoniot Jump Condition Lax Admissibility Conditions Asymptotic Behavior of Hugoniot Loci Centered Rarefactions Riemann Problems Riemann Problem for Linear Systems Riemann Problem for Shallow Water Entropy Functions Upwind Schemes Case J Case! Case! Lax-Friedrichs Scheme Rusanov Scheme Godunov Scheme Study: Maxwell's Equations Conservation Laws Characteristic Analysis Study: Gas Dynamics Conservation Laws Thermodynamics Characteristic Analysis Entropy Function Centered Rarefaction Curves Jump Conditions Riemann Problem Reflecting Walls Study: Magnetohydrodynamics (MHD) Conservation Laws Characteristic Analysis Entropy Function Centered Rarefaction Curves Jump Conditions ix

5 Contents 4.6 Case Study: Finite Deformation in Elastic Solids Eulerian Formulation of Equations of Motion for Solids Lagrangian Formulation of Equations of Motion for Solids Constitutive Laws Conservation Form of the Equations of Motion for Solids Jump Conditions for Isothermal Solids Characteristic Analysis for Solids Case Study: Linear Elasticity Case Study: Vibrating String Conservation Laws Characteristic Analysis Jump Conditions Lax Admissibility Conditions Entropy Function Wave Families for Concave Tension Wave Family Intersections Riemann Problem Solution Case Study: Plasticity Lagrangian Equations of Motion Constitutive Laws Centered Rarefactions Hugoniot Loci Entropy Function Riemann Problem Case Study: Polymer Model Constitutive Laws Characteristic Analysis Jump Conditions Riemann Problem Solution Case Study: Three-Phase Buckley-Leverett Flow Constitutive Models Characteristic Analysis Umbilic Point Elliptic Regions Case Study: Schaeffer-Schechter-Shearer System Approximate Riemann Solvers Design of Approximate Riemann Solvers Artificial Diffusion Rusanov Solver Weak Wave Riemann Solver 293

6 Contents XI Colella-Glaz Riemann Solver Osher-Solomon Riemann Solver Bell-Colella-Trangenstein Approximate Riemann Problem Solver Roe Riemann Solver Harten-Hyman Modification of the Roe Solver Harten-Lax-van Leer Scheme HLL Solvers with Two Intermediate States Approximate Riemann Solver Recommendations 319 Methods for Scalar Laws Convergence Consistency and Order Linear Methods and Stability Convergence of Linear Methods Entropy Conditions and Difference Approximations Bounded Convergence Monotone Schemes Nonlinear Stability Total Variation Total Variation Stability Other Stability Notions Propagation of Numerical Discontinuities Monotonie Schemes Smoothness Monitor Monotonizations MUSCL Scheme Discrete Entropy Conditions E-Schemes Total Variation Diminishing Schemes Sufficient Conditions for Diminishing Total Variation Higher-Order TVD Schemes for Linear Advection Extension to Nonlinear Scalar Conservation Laws Slope-Limiter Schemes Exact Integration for Constant Velocity Piecewise Linear Reconstruction Temporal Quadrature for Flux Integrals Characteristic Tracing Flux Evaluation Non-Reflecting Boundaries with the MUSCL Scheme 390

7 Xll Contents 5.10 Wave Propagation Slope Limiter Schemes Cell-Centered Wave Propagation Side-Centered Wave Propagation Higher-Order Extensions of the Lax-Friedrichs Scheme Piecewise Parabolic Method Essentially Non-Oscillatory Schemes Discontinuous Galerkin Methods Weak Formulation Basis Functions Numerical Quadrature Initial Data Limiters Timestep Selection Case Studies Case Study: Linear Advection Case Study: Burgers' Equation Case Study: Traffic Flow Case Study: Buckley-Leverett Model 427 Methods for Hyperbolic Systems First-Order Schemes for Nonlinear Systems Lax-Friedrichs Method Random Choice Method Godunov's Method Godunov's Method with the Rusanov Flux Godunov' s Method with the Harten-Lax-vanLeer (HLL) Solver Godunov's Method with the Harten-Hyman Fix for Roe's Solver Second-Order Schemes for Nonlinear Systems Lax-Wendroff Method MacCormack's Method Higher-Order Lax-Friedrichs Schemes TVD Methods MUSCL Wave Propagation Methods PPM ENO Discontinuous Galerkin Method 453

8 Contents 6.3 Case! Studies Wave Equation Shallow Water Gas Dynamics MHD Nonlinear Elasticity Cristescu's Vibrating String Plasticity Polymer Model Schaeffer-Schechter-Shearer Model Methods in Multiple Dimensions Numerical Methods in Two Dimensions Operator Splitting Donor Cell Methods Traditional Donor Cell Upwind Method First-Order Corner Transport Upwind Method Wave Propagation Form of First-Order Corner Transport Upwind Second-Order Corner Transport Upwind Method Wave Propagation D Lax-Friedrichs First-Order Lax-Friedrichs Second-Order Lax-Friedrichs Multidimensional ENO Discontinuous Galerkin Method on Rectangles Riemann Problems in Two Dimensions Burgers' Equation Shallow Water Gas Dynamics Numerical Methods in Three Dimensions Operator Splitting Donor Cell Methods Corner Transport Upwind Scheme Linear Advection with Positive Velocity Linear Advection with Arbitrary Velocity General Nonlinear Problems Second-Order Corner Transport Upwind Wave Propagation 521 хш

9 XIV Contents 7.4 Curvilinear Coordinates Coordinate Transformations Spherical Coordinates Case Study: Eulerian Gas Dynamics in Spherical Coordinates Case Study: Lagrangian Solid Mechanics in Spherical Coordinates Cylindrical Coordinates Case Study: Eulerian Gas Dynamics in Cylindrical Coordinates Case Study: Lagrangian Solid Mechanics in Cylindrical Coordinates Source Terms Geometric Flexibility Adaptive Mesh Refinement Localized Phenomena Basic Assumptions Outline of the Algorithm Timestep Selection Advancing the Patches Boundary Data Flux Computation Time Integration Regridding Proper Nesting Tagging Cells for Refinement Tag Buffering Logically Rectangular Organization Initializing Data after Regridding Refluxing Upscaling Initialization Object Oriented Programming Programming Languages AMR Classes Geometric Indices Boxes Data Pointers Lists 569

10 FlowVanables Timesteps TagBoxes DataBoxes EOSModels Patch Level ScalarLaw Example ScalarLaw Constructor initialize stabledt stuffmodelghost stuffboxghost computefluxes conservativedifference finderrorcells Numerical Example Linear Elasticity Example Gas Dynamics Examples Contents xv Bibliography Index