Preliminary results on very high order oscillation free Residual Distribution schemes for hyperbolic systems

Since a few years, the Residual Distribution (RD) methodology has become a more mature numerical technique. It has been applied to several physical problems : standard aerodynamic problems, MHD flows, multiphase flow problems, the Shallow Water equations, aeroacoustic, to give a few examples [1,2,3]. There are also interesting contributions from other groups. These schemes are devoted to steady and unsteady problems and lead to nonlinear problems. In most cases, the expected order of accuracy is second order in space (and time for non steady problems). The accuracy is obtained only if the solution of the non linear problem is obtained with enough accuracy. The RD schemes borrow features from the finite element world : they can be interpreted as a Petrov– Galerkin scheme where the test space may depend on the solution. They also borrow features from the high order TVD–like world in that the stabilization mechanism for non smooth problems is constructed by mimicking the non oscillatory mechanism of some monotone schemes without sacrificing (formal) accuracy. Since some times there is an attempt to extend them to more than second order accuracy. Early results were obtained in [4] for steady scalar problems. The expected order of accuracy (third and fourth order) was obtained in practice for smooth solutions, while the schemes were proved to be non oscillatory. However, some problems were observed by M. Ricchiuto in his PhD thesis and by M. Hubbards (Leeds). More recently, the authors have adapted the stabilization method of [5] to construct schemes that are also stable, non oscillatory, exempt from the criticism of Ricchiuto and Hubbard and reach the expected order on smooth problems, i.e. third and fourth order. Some interesting results are also obtained by the Bari group (de Palma et al., Italy) In the talk, we will review these methods, show how to extend them to systems. Preliminary examples taken from the Cauchy Riemann equations and some simple steady Euler problems will be presented during the meeting.