Positivity preserving and entropy consistent approximate Riemann solvers dedicated to the high-order MOOD-based Finite Volume discretization of Lagrangian and Eulerian gas dynamics

Abstract

In this paper we propose to revisit the notion of simple Riemann solver both in Lagrangian and Eulerian coordinates following the seminal work of Gallice ”Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates” in Numer. Math., 94, 2003. We provide in this work the relation between the Eulerian and Lagrangian forms of systems of conservation laws in 1D. Then an approximate (simple) Lagrangian Riemann solver for gas dynamics is derived based on the notions of positivity preservation and entropy control. Its Eulerian counterpart is further deduced. Next we build the associated 1D first-order accurate cell-centered Lagrangian Godunov-type Finite Volume scheme and show numerically its behaviors on classical test cases. Then using the Lagrangian–Eulerian relationships, we derive and test the Eulerian Godunov-type Finite Volume scheme, which inherits by construction the properties of the Lagrangian solver in terms of positivity preservation and well-defined CFL condition. At last we extend this Eulerian scheme to arbitrary orders of accuracy using a Runge–Kutta time discretization, polynomial reconstruction and an a posteriori MOOD limiting strategy. Numerical tests are carried out to assess the robustness, accuracy, and essentially non-oscillatory properties of the numerical methods.