Recasting an operator splitting solver into a standard finite volume flux-based algorithm. The case of a Lagrange-Projection-type method for gas dynamics

TL;DR

This paper introduces a modified operator splitting scheme for compressible flow simulation that transforms into a flux-splitting method, offering improved computational efficiency, stability, and extensibility to multi-dimensional and high-order problems.

Contribution

It presents a simple modification of an existing Lagrange-projection method, converting it into a flux-splitting scheme with enhanced efficiency, stability, and flexibility for complex flow simulations.

Findings

The new scheme is less computationally expensive and more memory efficient.

It preserves positivity of mass, energy, and entropy under CFL conditions.

The method is effective for highly compressible and low-Mach flows in 1D and 2D.

Abstract

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes.