Towards Very High Order Godunov Schemes

Abstract

We present an approach, called ADER, for constructing non-oscillatory advection schemes of very high order of accuracy in space and time; the schemes are explicit, one step and have optimal stability condition for one and multiple space dimensions. The approach relies on essentially non-oscillatory reconstructions of the data and the solution of a generalised Riemann problem via solutions of derivative Riemann problems. The schemes may thus be viewed as Godunov methods of very high order of accuracy. We present the ADER formulation for the linear advection equation with constant coefficients, in one and multiple space dimensions. Some preliminary ideas for extending the approach to non-linear problems are also discussed. Numerical results for one and two-dimensional problems using schemes of upto 10-th order accuracy are presented.