Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.

On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws

The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.

Restoration of the contact surface in the HLL-Riemann solver

P. L. RoeCollege of Aeronautics, Cranfield Institute of Technology,Cranfield MK43 0AL, EnglandIntroductionComputer simulations of fluid flow provide today the sort of detailedinformation concerning special cases that could previously only beobtained from experime.nts. The computer is attractive as a replacement forexperiments that are difficult, dangerous, or expensive, and as an alternativeto experiments that are impossible. Nevertheless, a computer simulationdoes not have quite the same status as a physical experiment because atpresent there usually remains some doubt about its accuracy. Even thoughthe computer code may be free of error to the extent that it operates exactlyas its author intended, it is seldom possible to give a rigorous proof thatthese intentions were in all respects correct. Most of the practical codeswritten to solve complicated problems contain empirical features, some-times in the form of "adjustable constants" whose values must be "tuned"by appeal to the experiments that the simulations are intended to displace.A computer code is described as being "robust" if it has the virtue ofgiving reliable answers to a wide range of problems without needing to beretuned. The ideal code would be one that fully met some declaredspecification of accuracy and problem range, and whose every line was anecessary contribution to that aim. Few codes yet approach that ideal; amajor impediment is that we presently have little idea what properties canbe specified without contradiction.In recent years, however, our understanding of computer codes for aparticular class of problems has advanced some way toward completeness.The problems are sufficiently complex that naive numerical techniques canproduce disaster, yet sufficiently simple that well-understood physics can3370066-4189/86/0115-0337502.00www.annualreviews.org/aronline Annual Reviews

Characteristic-based schemes for the euler equations

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.

https://hal.archives-ouvertes.fr/hal-01620642/document

Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics

Numerical solution of multidimensional problems of gas dynamics

I: Elementary Properties of Deformations and Stresses. Deformations and Strain. General Properties of Stresses. Stressed State at a Point. Isotropic Elastic Medium, Equation of State, and the Murnaghan Formulas. Viscous Stresses in an Isotropic Medium. Strain Tensors and Distortion Tensor. Murnaghan Formulas in the Tensor Form and Small Deformations. Moving Medium. Murnaghan Formulas. II: Effective Elastic Deformation. Relaxation of Stresses. Equations for the Metric Effective Elastic Deformation Tensor. Compatibility Conditions. Description of Relaxation of Shear Stresses by Equations for Effective Distortion. Defects of Crystal Lattice. Review of Notions. Dislocations. III: Differential Equations of Dynamical Processes. Conservation Laws. Consequences of Conservation Laws. Deformation of a Layer and a Bar. Equations for One-Dimensional Nonstationary Processes and Structure of Stationary Waves in a Maxwell Medium. IV: Well Posedness of Differential Equations and Thermodynamics. Well Posedness Conditions for Equations of Elasticity. Equations of Linear Elasticity as Symmetric Hyperbolic Equations. Symmetrization of Equations of Gas Dynamics. Influence of Small Viscosity on the Behavior of Solutions. Well Posedness Stability and Postulates of Phenomenological Thermodynamics. V: Multi-Dimensional Thermodynamically Compatible Conservation Laws. Gas Dynamics Equations and Magnetic Hydrodynamics Equations. Equations of Elasticity. Structure of Multi Dimensional Equations. Symmetric Hyperbolicity in the Three-Dimensional Case. Appendix: Structure of Thermodynamically Compatible Systems S.K. Godunov. Mathematical Aspects. The Simplest Galilei-Invariant Thermodynamically Compatible Systems. Methods of Constructing Equations. Some Facts of the Theory of Representations of Orthogonal Transformations. The Clebsch-Gordan Coefficients. Orthogonal Invariants. Literature. Subject Index.

Elements of Continuum Mechanics and Conservation Laws

Introduction: Euclidean linear spaces Orthogonal and unitary linear transformations Orthogonal and unitary transformations. Singular values Matrices of operators in the Euclidean space: Unitary similar transformations. The Schur theorem Alternation theorems The Weyl inequalities Variational principles Resolvent and dichotomy of spectrum Quadratic forms in the spectrum dichotomy problem Matrix equations and projections The Hausdorff set of a matrix Application of spectral analysis. The most important algorithms: Matrix operators as models of differential operators Application of the theory of functions of complex variable Computational algorithms of spectral analysis Bibliography Index.

S. K. Godunov

Papers

Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics

Numerical solution of multidimensional problems of gas dynamics

Elements of Continuum Mechanics and Conservation Laws

Modern Aspects of Linear Algebra

Nonstationary equations of nonlinear elasticity theory in eulerian coordinates