Speaker：Raphaël Loubère（University of Bordeaux）

Time：2023-12-15, 10:20

Location：Conference Room 105 at Experiment Building at Haiyun Campus

Abstract:

In this talk, we propose to reuse the notion of simple Riemann solver in Lagrangian coordinates to develop a new Eulerian Finite Volume (FV) scheme in the multi-dimensional case on unstructured meshes. As a proof of concept, we entirely derive the associated first-order accurate cell-centered Eulerian scheme for compressible flows using the Lagrangian to Eulerian correspondence. First, the Lagrangian simple Riemann solver is used as a building block to construct its Eulerian counterpart. This solver inherits by construction the properties of the Lagrangian one, mainly: positivity preservation, entropy dissipation, well-defined CFL condition and wave-speed ordering. From this Riemann solver, a classical two-point first-order Finite Volume Eulerian scheme can be deduced for which the numerical fluxes of a given cell are computed only with respect to two neighbors through a common face. Next, we introduce another Eulerian numerical scheme which involves a multi-dimensional Lagrangian nodal solver, leading to the so-called multi-point Riemann solver that involves all surrounding cells, including corner cells. The conservation is no more relying on a one-to-one flux cancellation across a face like most FV approaches. Conversely, in this work, conservation is retrieved on a node basis. An associated first-order Eulerian scheme is derived on the basis of this multi-point nodal-based Riemann solver. We prove that this FV multi-point scheme still inherits some good properties with the extra-property of coupling all neighbor cells in a consistent way. A set of numerical results on general 2D unstructured grids are presented on several classical two-dimensional test cases, showing that the two-point scheme generates spurious instabilities such as the infamous carbuncle phenomena, while the multi-point scheme seems insusceptible to those.