Synopses & Reviews
This edited review book on Godunov methods contains 97 articles, all of which were presented at the international conference on Godunov Methods: Theory and Applications, held at Oxford, in October 1999, to commemorate the 70th birthday of the Russian mathematician Sergei K. Godunov. The central theme of this book is numerical methods for hyperbolic conservation laws following Godunov's key ideas contained in his celebrated paper of 1959. Hyperbolic conservation laws play a central role in mathematical modelling in several distinct disciplines of science and technology. Application areas include compressible, single (and multiple) fluid dynamics, shock waves, meteorology, elasticity, magnetohydrodynamics, relativity, and many others. The successes in the design and application of new and improved numerical methods of the Godunov type for hyperbolic conservation laws in the last twenty years have made a dramatic impact in these application areas. The 97 papers cover a very wide range of topics, such as design and analysis of numerical schemes, applications to compressible and incompressible fluid dynamics, multi-phase flows, combustion problems, astrophysics, environmental fluid dynamics, and detonation waves. This book will be a reference book on the subject of numerical methods for hyperbolic partial differential equations for many years to come. All contributions are self-contained but do contain a review element. There is a key paper by Peter Sweby in which a general overview of Godunov methods is given. This contribution is particularly suitable for beginners on the subject. This book is unique: it contains virtually everything concerned with Godunov-type methods for conservation laws. As such it will be of particular interest to academics (applied mathematicians, numerical analysts, engineers, environmental scientists, physicists, and astrophysicists) involved in research on numerical methods for partial differential equations; scientists and engineers concerned with new numerical methods and applications to scientific and engineering problems e.g., mechanical engineers, aeronautical engineers, meteorologists; and academics involved in teaching numerical methods for partial differential equations at the postgraduate level.
Table of Contents
Oleinik's E-Condition from the Viewpoint of Numerics; H. Aiso. On Some New Results for Residual Distribution Schemes; R. Abgrall, T.J. Barth. Simulations of Relativistic Jets with Genesis; M.A. Aloy, et al. Relativistic Jets from Collapsars; M.A. Aloy, et al. Exact Computation in Numerical Linear Algebra: The Discrete Fourier Transform; J.A.D.W. Anderson, P.K. Sweby. Comparative Study of HLL, HLLC and Hybrid Riemann Solvers in Unsteady Compressible Flows; A. Bagabir, D. Drikakis. A New Reconstruction Technique for the Euler Equations of Gas Dynamics with Source Terms; P. Bartsch, A. Borzi. Colella-Glaz Splitting Scheme for Thermally Perfect Gases; A. Beccantini. Meshless Particle Methods: Recent Developments for Nonlinear Conservation Laws in Bounded Domain; B.B. Moussa. Application of Wave-propagation Algorithm to Two-dimensional Thermoelastic Wave Propagation in Inhomogeneous Media; A. Berezovski, G.A. Maugin. Unstructured Mesh Solvers for Hyperbolic PDEs with Source Terms: Error Estimates and Mesh Quality; M. Berzins, L.J.K. Durbeck. Constancy Preserving, Conservative Methods for Free-surface Models; L. Bonaventura, E. Gross. Hyperbolic-elliptic Splitting for the Pseudo-compressible Euler Equations; A. Bonfiglioli. Godunov Solution of Shallow Water Equations on Curvilinear and Quadtree Grids; A.G.L. Borthwick, et al. A High-order-accurate Reconstruction for the Computation of Compressible Flows on Cell-vertex Triangular Grids; L.A. Catalano. Numerical Experiments with Multilevel Schemes for Conservation Laws; G. Chiavassa, R. Donat. Volume-of-fluid Methods for Partial Differential Equations; P. Colella. Some New Godunov and Related Relaxation Methods for Two-phase Flow Problems; F. Coquel, et al. Development of Genuinely Multi-dimensional Upwind Residual Distribution Schemes for the System of Eight Wave Ideal Magnetohydrodynamic Equations on Uncunstructured Grids; A. Csík, et al. Application of TVD High Resolution Schemes to the Viscous Shock Tube Problem; V. Daru, C. Tenaud. Comparison of Numerical Solvers with Godunov Scheme for Multicomponent Turbulent Flows; E. Declercq. Godunov-type Schemes for the MHD Equations; A. Dedner, et al. Absorbing Boundary Conditions for Astrophysical MHD Simulations; A. Dedner, et al. About Kinetic Schemes Built in Axisymmetrical and Spherical Geometries; S. Dellacherie. Lagrangian Systems of Conservation Laws and Approximate Riemann Solvers; B. Després. Intermediate Shocks in 3D MHD Bow Shock Flows; H. De Sterck, S. Poedts. A Second Order Godunov-type Scheme for Naval Hydrodynamics; A. Di Mascio, et al. Uniformly High-order Methods for Unsteady Incompressible Flows; D. Drikakis. Application of the Finite Volume Method with Osher Scheme and Split Technique for Different Types of Flow in a Channel; K.S. Erduran, V. Kutija. A-priori Estimates for a Semi-Lagrangian Scheme for the Wave Equation; M. Falcone, R. Ferretti. Interstellar Shock Structures in Weakly Ionised Gases; S.A.G.E. Falle. The Ghost Fluid Method for Numerical Treatment of Discontinuities and Interfaces; R.P. Fedkiw. A Hybrid Primitive-Conservative Upwind Scheme for the Drift Flux Model; K.K. Fjelde, K.H. Karlsen. Numerical Simulations of Relativistic Wind Accretion onto Black Holes Using Godunov-type Methods; J.A. Font, et al. A Second Order Accurate, Space-time Limited, BDF Scheme for the Linear Advection Equation; S.A. Forth. Multidimensional Upwind Schemes: Application to Hydraulics; P. Garcia-Navarro, et al. HELMIT &endash; A New Interface Reconstruction Algorithm; R. Giddings. A Godunov-type Method for Studying the Linearised Stability of a Flow. Application to the Richtmyer-Meshkov Instability; E. Godlewski, et al. Thermodynamics, Conservation Laws and their Rotation Invariance; S.K. Godunov. A New Limiter that Improves TVD-MUSCL Schemes; L. Gozalo, R. Abgrall. Exact Roe Linearisation for van der Waals' Gas; A. Guardone, L. Quartapelle. The Godunov-Ryabenkii Condition: The Beginning of a New Stability Theory; B. Gustafsson. A Front Tracking Method for Hybrid Grids; D. Hänel, et al. A Problem of Classical Shock Capturing Finite Volume Schemes in Hypersonic Flows; V. Hannemann. Orientation Effects on Bent Extragalactic Jets; S. Higgins, et al. Operator Splitting for Convection-dominated Nonlinear Partial Differential Equations; H. Holden, et al. Balancing Source Terms and Flux Gradients in Finite Volume Schems; M.E. Hubbard, P. Garcia-Navarro. Riemann Solvers in General Relativistic Hydrodynamics; J.M. Ibáñyez, et al. A Fully Adaptive Multiresolution Scheme for Shock Computations; M.K. Kaibara, S.M. Gomes. Application of a Godunov-type ALE-method to Underwater Shock Waves; A. Klomfass, et al. Numerical Simulation of 2-D Two-phase Flows with Interface; S. Kokh, G. Allaire. Relativistic MHD Simulations Using a Godunov-type Method; S. Komissarov. Godunov Type Methods on Unstructured Grids and Local Mesh Refinement; D. Kröner, T. Gessner. 3D Visualization of Shock Waves Using Volume Rendering; J.O. Langseth. Gas Flows Generated by Propellant Burning; C.A. Lowe, J.F. Clarke. Finite Volume Evolution Galerkin Methods for Multidimensional Hyperbolic Systems; M. Lukáčová-Medvidová, et al. The Numerical Simulation of Relativistic Fluid Flow with Strong Shocks; A. Marquina. An Artificial Compression Procedure Via Flux Correction; V. Martínez. A Second-order Time-splitting Technique for Advection-Dispersion Equation on Unstructured Grids; A. Mazzia, et al. Towards Implicit Godunov Method: Exact Linearisation of the Numerical Flux; I. Men'Shov, Y. Nakamura. Mass Flux Computations as a Key to the Carbuncle Phenomenon; J.-M. Moschetta. On the Positivity of FVS Schemes; J.-M. Moschetta, J. Gressier. The Carbuncle Phenomenon: a Genuine Euler Instability? J.-M. Moschetta, et al. A Godunov-type Solver for the Maxwell Equations with Divergence Cleaning; C.-D. Munz, et al. Convergence of Kinetic Approximation to Nonlinear Parabolic Problems; G. Naldi, et al. On Options for the Numerical Modelling of the Diffusion Term in River Pollution Simulations; S. Neelz, et al. Multidimensional Flux-vector-splitting and High-resolution Characteristic Schemes; S. Noelle. A Comparison of Roe, VFFC and AUSM+ Schemes for Two-phase Water/Steam Flows; H. Peillere, et al. Low Dissipation Entropy Fix for Positivity Preserving Roe's Scheme; M. Pelanti, et al. Bicharacteristic Methods for Multidimensional Hyperbolic Systems; M.H. Pham, et al. An exact Riemann Solver for Multidimensional Special Relativistic Hydrodynamics; J.A. Pons, et al. Experience with the Osher Scheme for Applied Aerodynamics; N. Qin. A High-resolution Godunov Method for Modelling Anomalous Fluid Behaviour; W.J. Rider, J.W. Bates. Towards Godunov-type Methods for Hyperbolic Conservation Laws with Stiff Relaxation; P.L. Roe, J.A.F. Hittinger. Thermodynamics and Hyperbolic Systems of Balance Laws in Continuum Mechanics; E.I. Romensky. Development and Application of High-resolution Adaptive Numerical Techniques in Shock Wave Research Center; T. Saito, et al. Interfaces, Detonation Waves, Cavitation and the Multi-phase Godunov Method; R. Saurel. One-dimensional Calculation of Unsteady Open Channel Flow Using Adaptive Mesh Refinement; J. Schramm, et al.