An Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Scheme for Compressible Multi-Material Flows on Adaptive Quadrilateral Meshes (Communications in Computational Physics, 2024)
In this paper, a direct arbitrary Lagrangian-Eulerian (ALE) discontinuousGalerkin (DG) scheme is proposed for simulating compressible multi-material flowson the adaptive quadrilateral meshes. Our scheme couples a conservative equationrelated to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture. The coupled system is discretized in the reference elementand we use a kind of Taylor expansion basis functions to construct the interpolationpolynomials of the variables. We show the property that the material derivatives ofthe basis functions in the DG discretization are equal to zero, with which the schemeis simplified. In addition, the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from [H.Z. Tang and T. Tang, Adaptive mesh methodsfor one-and two-dimensional hyperbolic conservation laws, SIAM J. NUMER. ANAL].This adaptive mesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation nearthe material interfaces in the simulations. With the help of this adaptive mesh method,the resolution of the solution near the target regions can be greatly improved and thecomputational efficiency of the simulation is increased. Our scheme can be applied inthe simulations for the gas and water media efficiently, and it is more concise comparedto some other methods such as the indirect ALE methods. Several examples includingthe gas-water flow problem are presented to demonstrate the efficiency of our scheme, and the results show that our scheme can capture the wave structures sharply withhigh robustness.