High-order ADER Discontinuous Galerkin schemes for a symmetric hyperbolic model of compressible barotropic two-fluid flows

TL;DR

This paper develops a high-order ADER discontinuous Galerkin method for a symmetric hyperbolic model of compressible two-fluid flows, addressing hyperbolicity issues and demonstrating high accuracy through various test cases.

Contribution

It introduces a high-order DG scheme for a hyperbolic two-fluid model, including methods to restore hyperbolicity and extensive validation against benchmarks.

Findings

The model is only weakly hyperbolic in multiple dimensions.

Two methods successfully restore strong hyperbolicity.

The scheme achieves high-order accuracy and good agreement with reference solutions.

Abstract

This paper presents a high-order discontinuous Galerkin finite element method to solve the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flow, introduced by Romenski et al., in multiple space dimensions. In the absence of algebraic source terms, the model is endowed with a curl constraint on the relative velocity field. In this paper, the hyperbolicity of the system is studied for the first time in the multidimensional case, showing that the original model is only weakly hyperbolic in multiple space dimensions. To restore strong hyperbolicity, two different methodologies are used: i) the explicit symmetrization of the system, which can be achieved by adding terms that contain linear combinations of the curl involution, similar to the Godunov-Powell terms in the MHD equations; ii) the use of the hyperbolic generalized Lagrangian multiplier (GLM) curl-cleaning approach forwarded. The PDE system is solved using a high-order ADER discontinuous Galerkin method with a posteriori sub-cell finite volume limiter to deal with shock waves and the steep gradients in the volume fraction commonly appearing in the solutions of this type of model. To illustrate the performance of the method, several different test cases and benchmark problems have been run, showing the high-order of the scheme and the good agreement when compared to reference solutions computed with other well-known methods.