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

A class of high-resolution algorithms is developed for advection of a scalar quantity in a given incompressible flow field in one, two, or three space dimensions. Multidimensional transport is modeled using a wave-propagation approach in which the flux at each cell interface is built up on the basis of information propagating in the direction of this interface from neighboring cells. A high-resolution second-order method using slope limiters is quite easy to implement. For constant flow, a minor modification gives a third-order accurate method. These methods are stable for Courant numbers up to 1. Fortran implementations are available by anonymous ftp.

High-resolution conservative algorithms for advection in incompressible flow

ALE formulation for fluid–structure interaction problems

Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows

https://www.math.u-bordeaux.fr/~rabgrall/mes_papiers/JCP_167_2_2001.pdf

Toward the ultimate conservative scheme: following the quest

In this paper the optimal linear, positive schemes for constant-coefficient advection in two or three dimensions are presented. These are the generalizations of first-order upwinding in one dimension. By comparison with a dimension-by-dimension treatment the optimum schemes have much lower numerical diffusion, and permit larger timesteps.

P. L. Roe

Papers

Optimum positive linear schemes for advection in two and three dimensions