A second-order accurate and robust numerical scheme is developed for the Kapila model to simulate compressible multiphase flows. The scheme is formulated within the temporal-spatial coupling framework with the generalized Riemann problem (GRP) solver applied as the cornerstone. The use of the GRP solver enhances the capability of the resulting scheme to handle the stiffness of the Kapila model in two ways. Firstly, in addition to Riemann solutions, the time derivatives of flow variables at cell interfaces are obtained by the GRP solver. The coupled values, i.e. Riemann solutions and time derivatives, lead to a straightforward approximation to the velocity divergence at the next time level, enabling a semi-implicit time discretization to the volume fraction equation. Secondly, the use of time derivatives enables numerical fluxes to comprehensively account for the effect of the source term, which includes interactions between phases. The robustness of the resulting numerical scheme is therefore further improved. Several challenging numerical experiments are conducted to demonstrate the performance of the proposed finite volume scheme. In particular, a test case with a nonlinear smooth solution is designed to verify the numerical accuracy.

翻译：针对Kapila模型，发展了一种二阶精度且鲁棒的数值格式，用于模拟可压缩多相流动。该格式在时空耦合框架下构建，以广义黎曼问题求解器为核心。GRP求解器的应用从两方面增强了格式处理Kapila模型刚性的能力：首先，除黎曼解外，GRP求解器还提供了单元界面处流动变量的时间导数。这些耦合值（即黎曼解与时间导数）可实现下一时间步速度散度的直接近似，从而对体积分数方程采用半隐式时间离散；其次，利用时间导数使数值通量能够全面考虑源项（包括相间相互作用）的影响，进一步提高了数值格式的鲁棒性。通过多个具有挑战性的数值实验验证了所提有限体积格式的性能，特别设计了一个包含非线性光滑解的算例以检验数值精度。