Runge-Kutta residual distribution schemes

We are concerned with the solution of time-dependent nonlinear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge-Kutta-type time stepping (discretisation in time). The introduced nonlinear blending procedure allows us to retain the explicit character of the time stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. Key-words: Hyperbolic Conservation Laws, Time-dependent problems, Second order schemes, Residual Distribution, Runge-Kutta time-stepping ∗ School of Computing, University of Leeds, UK † leeds ‡ Inria Bordeaux Sud-Ouest ha l-0 08 65 15 4, v er si on 1 24 S ep 2 01 3 Runge-Kutta Residual Distribution Schemes Résumé : On considère l’approximation de solution de lois de conservation hyperboliques avec une combinaison des mèthodes de type Reisdual Distribution avec des schémas Runge-Kutta en temps. On propose une construction de schéma non-linéaire de type Blended qui ne nécéssite pas la résolution d’un systéme non-linéaire, donc permettan de retenir le caractére explicite de la mèthode. L’approche proposée est validée sur des nombreaux cas test. Mots-clés : Lois de conservation Hyperboliques, problèmes instationnaires, schémas d’ordre deux, Residual Distribution, Runge-Kutta ha l-0 08 65 15 4, v er si on 1 24 S ep 2 01 3 Runge-Kutta Residual Distribution Schemes 3