Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: Quasi-conservative formulation with subcell finite volume corrections

Abstract

We present a novel quasi-conservative arbitrary high order accurate ADER (Arbitrary-Derivative) discontinuous Galerkin method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax–Wendroff theorem. Within this formalism, we also reformulate classical results concerning smooth solutions, contact discontinuities and moving interfaces. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax–Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics.To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude the presentation by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble for which we are able to avoid the development of any spurious oscillations.