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

TL;DR

This work addresses robustly simulating barotropic compressible two-fluid flows within the SHTC framework by deploying a high-order ADER-DG scheme with a posteriori subcell limiting. The authors show that the multidimensional barotropic SHTC model is only weakly hyperbolic and propose two routes to restore strong hyperbolicity: GLM curl-cleaning and symmetric Powell-type terms. They implement a one-step ADER-DG discretization with a space–time predictor and MOOD-style limiter, and validate it through accuracy tests, 1D Riemann problems, 2D explosion problems, and a dambreak scenario, achieving high-order convergence and excellent agreement with reference solutions. The results demonstrate a robust, high-fidelity framework for diffuse-interface two-fluid flows and pave the way for extensions to more phases and curl-free formulations with potential applications in engineering and physics.

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.

Figures (5)

• Figure 1: 1D Riemann problem solved with a fourth-order ADER-DG scheme with TVD subcell limiter on a Cartesian mesh at time $t=0.25$. Top row: densities of each phase, $\rho_1$ and $\rho_2$. Second row: mixture density $\rho$ and $\alpha$. Third row: velocities $u_1$ and $u_2$. Bottom row: mixture velocity $u$ (left) and relative velocity $w=u_1-u_2$ (right).
• Figure 2: 2D circular explosion problem for initial condition CE1 in \ref{['tab:CE']} solved on a Cartesian mesh at time $t=0.1$, in comparison with the radial reference solution. Top row: densities of each phase, $\rho_1$ and $\rho_2$. Second row: mixture density $\rho$ and $\alpha$. Third row: velocities $u_1$ and $u_2$. Bottom row: mixture velocity $u$ (left) and relative velocity $w=u_1-u_2$ (right).
• Figure 3: 2D circular explosion problem for initial condition CE2 in \ref{['tab:CE']} solved on a Cartesian mesh at time $t=0.2$, compared with the radial reference solution. Top row: densities of each phase, $\rho_1$ and $\rho_2$. Second row: mixture density $\rho$ and $\alpha$. Third row: velocities $u_1$ and $u_2$. Bottom row: mixture velocity $u$ (left) and relative velocity $w=u_1-u_2$ (right).
• Figure 4: Left: Limiter map of the explosion problem in 2D. The values in red mean that the limiter is activated. Right: 3D plot with the variable $\rho_2$ in the $z-$axis.
• Figure 5: Dambreak problem at time $t = 0.4$. Top: reference solution, computed with a third-order ADER-WENO finite volume scheme on a very fine uniform Cartesian grid, solving the inviscid and barotropic reduced Baer--Nunziato model presented in Dumbser2011TwoPhase. Center: Numerical solution, computed using an ADER-DG scheme with a posteriori sub-cell limiter, to solve the barotropic SHTC model proposed in this work. Bottom: Comparison of the free-surface profile obtained for both models.