Riemann Solvers and Numerical Methods for Fluid Dynamics

Transcription

1 Eleuterio R Toro Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introduction With 223 Figures Springer

2 Table of Contents Preface V 1. The Equations of Fluid Dynamics The Euler Equations Conservation-Law Form Other Compact Forms Thermodynamic Considerations Units of Measure Equations of State (EOS) Other Variables and Relations Ideal Gases Covolume and van der Waal Gases Viscous Stresses Heat Conduction Integral Form of the Equations Time Derivatives Conservation of Mass Conservation of Momentum Conservation of Energy Submodels \ Summary of the Equations Compressible Submodels Incompressible Submodels Notions on Hyperbolic Partial Differential Equations Quasi-Linear Equations: Basic Concepts The Linear Advection Equation Characteristics and the General Solution The Riemann Problem Linear Hyperbolic Systems Diagonalisation and Characteristic Variables The General Initial-Value Problem The Riemann Problem The Riemann Problem for Linearised Gas Dynamics.. 58

3 XII Table of Contents Some Useful Definitions Conservation Laws Integral Forms of Conservation Laws Non-Linearities and Shock Formation Characteristic Fields Elementary-Wave Solutions of the Riemann Problem Some Properties of the Euler Equations The One-Dimensional Euler Equations Conservative Formulation Non-Conservative Formulations Elementary Wave Solutions of the Riemann Problem Multi-Dimensional Euler Equations Two-Dimensional Equations in Conservative Form Three-Dimensional Equations in Conservative Form Three-Dimensional Primitive Variable Formulation The Split Three-Dimensional Riemann Problem Conservative Versus Non-Conservative Formulations Ill 4. The Riemann Problem for the Euler Equations Solution Strategy Equations for Pressure and Particle Velocity Function f L for a Left Shock Function / L for Left Rarefaction Function / R for a Right Shock Function /R for a Right Rarefaction Numerical Solution for Pressure Behaviour of the Pressure Function Iterative Scheme for Finding the Pressure Numerical Tests \ The Complete Solution Sampling the Solution Left Side of Contact: S = x/t<u* Right Side of Contact: S = x/t>u* The Riemann Problem in the Presence of Vacuum Case 1: Vacuum Right State Case 2: Vacuum Left State Case 3: Generation of Vacuum The Riemann Problem for Covolume Gases Solution for Pressure and Particle Velocity Numerical Solution for Pressure The Complete Solution Solution Inside Rarefactions The Split Multi-Dimensional Case FORTRAN 77 Program for Exact Riemann Solver 151

4 Table of Contents XIII 5. Notions on Numerical Methods Discretisation: Introductory Concepts Approximation to Derivatives Finite Difference Approximation to a PDE Selected Difference Schemes The First Order Upwind Scheme Other Well-Known Schemes Conservative Methods Basic Definitions Godunov's First-Order Upwind Method Godunov's Method for Burgers's Equation Conservative Form of Difference Schemes Upwind Schemes for Linear Systems The CIR Scheme Godunov's Method Sample Numerical Results Linear Advection The Inviscid Burgers Equation FORTRAN 77 Program for Godunov's Method The Method of Godunov for Non linear Systems Bases of Godunov's Method The Godunov Scheme Godunov's Method for the Euler Equations Evaluation of the Intercell Fluxes Time Step Size Boundary Conditions Numerical Results and Discussion Numerical Results for Godunov's Method Numerical Results from Other Methods Random Choice and Related Methods Introduction RCM on a Non-Staggered Grid The Scheme for Non-Linear Systems Boundary Conditions and the Time Step Size A Random Choice Scheme of the Lax-Friedrichs Type Review of the Lax-Friedrichs Scheme The Scheme The RCM on a Staggered Grid The Scheme for Non-Linear Systems A Deterministic First-Order Centred Scheme (FORCE) Analysis of the FORCE Scheme Random Numbers Van der Corput Pseudo-Random Numbers 234

5 XIV Table of Contents Statistical Properties Propagation of a Single Shock Numerical Results Concluding Remarks Flux Vector Splitting Methods Introduction The Flux Vector Splitting Approach Upwind Differencing The FVS Approach FVS for the Isothermal Equations Split Fluxes FVS Numerical Schemes FVS Applied to the Euler Equations Recalling the Equations The Steger-Warming Splitting The van Leer Splitting The Liou-Steffen Scheme Numerical Results Tests Results for Test Results for Test Results for Test Results for Test Approximate-State Riemann Solvers Introduction The Riemann Problem and the Godunov Flux Tangential Velocity Components Sonic Rarefactions Primitive Variable Riemann Solvers (PVRS) Approximations Based on the Exact Solver The Two-Rarefaction Riemann Solver (TRRS) A Two-Shock Riemann Solver (TSRS) Adaptive Riemann Solvers A PVRS-EXACT Adaptive Scheme A PVRS-TRRS-TSRS Adaptive Scheme Numerical Results The HLL and HLLC Riemann Solvers The Riemann Problem and the Godunov Flux The Riemann Problem and Integral Relations The HLL Approximate Riemann Solver The HLLC Approximate Riemann Solver Wave-Speed Estimates 302

6 Table of Contents XV Direct Wave Speed Estimates Pressure-Velocity Based Wave Speed Estimates Summary of the HLL and HLLC Riemann Solvers Contact Waves and Passive Scalars Numerical Results Closing Remarks and Extensions The Riemann Solver of Roe Bases of the Roe Approach The Exact Riemann Problem and the Godunov Flux.. 314,, Approximate Conservation Laws The Approximate Riemann Problem and the Intercell Flux The Original Roe Method The Isothermal Equations The Euler Equations The Roe-Pike Method The Approach The Isothermal Equations The Euler Equations An Entropy Fix The Entropy Problem The Harten-Hyman Entropy Fix The Speeds u», a tl, a* R Numerical Results and Discussion The Tests The Results Extensions The Riemann Solver of Osher Osher's Scheme for a General System Mathematical Bases Osher's Numerical Flux Osher's Flux for the Single-Wave Case Osher's Flux for the Inviscid Burgers Equation Osher's Flux for the General Case Osher's Flux for the Isothermal Equations Osher's Flux with P-Ordering Osher's Flux with O-Ordering Osher's Scheme for the Euler Equations Osher's Flux with P-Ordering Osher's Flux with O-Ordering Remarks on Path Orderings The Split Three-Dimensional Case Numerical Results and Discussion 372

7 XVI Table of Contents 12.5 Extensions High-Order and TVD Methods for Scalar Problems Introduction,., Basic Properties of Selected Schemes ; Selected Schemes Accuracy Stability WAF-Type High Order Schemes The Basic WAF Scheme Generalisations of the WAF Scheme MUSCL-Type High-Order Methods Data Reconstruction The MUSCL-Hancock Method (MHM) The Piece-Wise Linear Method (PLM) The Generalised Riemann Problem (GRP) Method MUSCL-Hancock Centred (SLIC) Schemes Other Approaches Semi-Discrete Schemes Implicit Methods Monotone Schemes and Accuracy Monotone Schemes A Motivating Example Monotone Schemes and Godunov's Theorem Spurious Oscillations and High Resolution Data Compatibility Total Variation Diminishing (TVD) Methods The Total Variation TVD and Monotonicity Preserving Schemes Flux Limiter Methods TVD Version of the WAF Method The General Flux-Limiter Approach TVD Upwind Flux Limiter Schemes TVD Centred Flux Limiter Schemes Slope Limiter Methods..' TVD Conditions Construction of TVD Slopes Slope Limiters Limited Slopes Obtained from Flux Limiters Extensions of TVD Methods TVD Schemes in the Presence of Source Terms TVD Schemes in the Presence of Diffusion Terms Numerical Results for Linear Advection 453

8 Table of Contents XVII 14. High-Order and TVD Schemes for Non-Linear Systems Introduction CFL and Boundary Conditions Weighted Average Flux (WAF) Schemes The Original Version of WAF A Weighted Average State Version Rarefactions in State Riemann Solvers TVD Version of WAF Schemes Riemann Solvers Summary of the WAF Method The MUSCL-Hancock Scheme The Basic Scheme A Variant of the Scheme TVD Version of the Scheme Summary of the MUSCL-Hancock Method Centred TVD Schemes Review of the FORCE Flux A Flux Limiter Centred (FLIC) Scheme A Slope Limiter Centred (SLIC) Scheme Primitive-Variable Schemes Formulation of the Equations and Primitive Schemes A WAF-Type Primitive Variable Scheme A MUSCL-Hancock Primitive Scheme Adaptive Primitive-Conservative Schemes Some Numerical Results Upwind TVD Methods Centred TVD Methods Splitting Schemes for PDEs with Source Terms Introduction Splitting for a Model Equation Numerical Methods Based on Splitting Model Equations Schemes for Systems Remarks on ODE Solvers First-Order Systems of ODEs Numerical Methods " Implementation Details for Split Schemes Concluding Remarks Methods for Multi-Dimensional PDEs Introduction Dimensional Splitting Splitting for a Model Problem Splitting Schemes for Two-Dimensional Systems 511

9 XVIII Table of Contents x Splitting Schemes for Three-Dimensional Systems Practical Implementation of Splitting Schemes in Three Dimensions Handling the Sweeps by a Single Subroutine Choice of Time Step Size..'.' The Intercell Flux and the TVD Condition Unsplit Finite Volume Methods Introductory Concepts Accuracy and Stability of Multidimensional Schemes A MUSCL-Hancock Finite Volume Scheme WAF-Type Finite Volume Schemes Two-Dimensional Linear Advection Three-Dimensional Linear Advection Schemes for Two-Dimensional Nonlinear Systems Schemes for Three-Dimensional Nonlinear Systems Non-Cartesian Geometries Introduction General Domains and Coordinate Transformation The Finite Volume Method for Non-Cartesian Domains Multidimensional Test Problems Explosions and Implosions Explosion Test in Two-Space Dimensions Implosions in Two Dimensions Explosion Test in Three Space Dimensions Shock Wave Reflection from Wedges Mach Number M s = 1.7 and tj> = 25 Degrees Mach Number M s = 1.2 and (j> = 30 Degrees Concluding Remarks Summary.; Extensions and Applications Shallow Water Equations Steady Supersonic Euler Equations The Artificial Compressibility Equations The Compressible Navier-Stokes Equations Compressible Materials Detonation Waves Multiphase Flows Magnetohydrodynamics (MHD) Relativistic Gas Dynamics Waves in Solids Teaching and Development Programs 567