High order space–time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems

Abstract

We present a class of high order finite volume schemes for the solution of non-conservative hyperbolic systems that combines the one-step ADER-WENO finite volume approach with space–time adaptive mesh refinement (AMR). The resulting algorithm, which is particularly well suited for the treatment of material interfaces in compressible multi-phase flows, is based on: (i) high order of accuracy in space obtained through WENO reconstruction, (ii) a high order one-step time discretization via a local space–time discontinuous Galerkin predictor method, and (iii) the use of a path conservative scheme for handling the non-conservative terms of the equations. The AMR property with time accurate local time stepping, which has been treated according to a cell-by-cell strategy, strongly relies on the high order one-step time discretization, which naturally allows a high order accurate and consistent computation of the jump terms at interfaces between elements using different time steps. The new scheme has been successfully validated on some test problems for the Baer–Nunziato model of compressible multiphase flows.