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.

翻译：本文提出一种高阶间断伽辽金有限元方法，用于求解由Romenski等人提出的可压缩两相流保守对称双曲热力学相容（SHTC）模型的正压版本，适用于多维空间。在无代数源项情况下，该模型对相对速度场施加了旋度约束。本文首次研究系统在多维情形下的双曲性，表明原始模型在多维空间中仅为弱双曲型。为恢复强双曲性，采用两种不同方法：i) 通过添加包含旋度对合线性组合的项实现系统的显式对称化，类似于磁流体动力学（MHD）方程中的Godunov-Powell项；ii) 采用双曲广义拉格朗日乘子（GLM）旋度清洁方法。该偏微分方程组采用高阶ADER间断伽辽金方法求解，并辅以后验子单元有限体积限制器处理激波以及此类模型解中常见的体积分数陡梯度。通过运行多个不同测试案例与基准问题来展示方法性能，结果表明该格式具有高阶精度，且与采用其他知名方法计算的参考解具有良好一致性。