A Non-dissipative Reconstruction Scheme for the Compressible Euler Equations

TL;DR

This paper introduces a conservative finite volume scheme for the Euler equations that accurately reconstructs shocks with minimal numerical diffusion, avoids spurious oscillations, and reduces wall heating effects.

Contribution

It develops a novel shock reconstruction scheme that is exact for pure shocks, exhibits low diffusion, and suppresses numerical artifacts in compressible flow simulations.

Findings

Exact for pure shocks, reproducing the true solution.

Low numerical diffusion with shocks spanning one or two cells.

Significantly reduces wall heating phenomena.

Abstract

We present a finite volume scheme, first on the Burgers equations, then on the Euler equations, based on a conservative reconstruction of shocks inside each cells of the mesh. Its main features are the following points. First, the scheme is exact whenever the initial datum is a pure shock, in the sense that the approximate solution is the exact solution averaged over the cells of the mesh. Second, the scheme has in general a very low numerical diffusion and the shocks have a width of one or two cells. Third, no spurious oscillations in the momentum appear behind slowly moving shocks, which is not the case in most of the scheme developed so far. We also present prospective result on the full Euler equations with energy. The wall heating phenomenon, which is an artificial elevation of the temperature when a shock reflects on a wall, is also drastically diminished.