An h-adaptive local discontinuous Galerkin method for second order wave equation: Applications for the underwater explosion shock hydrodynamics

Abstract

The underwater explosion (UNDEX) phenomenon involves hydrodynamic behaviors with a wide range of spatial scales, from macroscopic cavitation regions to refined areas with fluid-structure interactions around the shock wavefront. Correct description of the refined shock wavefront is important for determining the shock wave position and pressure peak. In this work, the local discontinuous Galerkin (LDG) method for the two-dimensional wave equation, which allows for the local triangular meshes refinement and merging during the shock wave propagation process, is presented to model the UNDEX shock hydrodynamics. This article is an extension of our previous work, where we discretize the UNDEX total wave formulation in the LDG framework and simulate the UNDEX shock wave and cavitation phenomenon. In order to save computational cost (including CPU time and computational storage) and capture the shock wavefront more accurately, the LDG method is extended to be in combination with the mesh adaption technique. The troubled-cell indicator, depending on the local jump of the pressure, is adopted to identify elements which require to be refined or coarsened. Typical numerical examples are tested to assess the capability of the present h-adaptive LDG method for capturing the UNDEX shock wave. The numerical results clearly demonstrate that the mesh adaption technique enables us to dynamically increase the mesh resolution in close proximity to the flow discontinuities with a large pressure jump. The local increase of the mesh resolution results in a substantial increase in the accuracy of the pressure peak. In terms of the computational efficiency, the h-adaptive LDG method can save nearly half of the calculation time in comparison to the non-adaptive LDG method on uniform meshes, whose mesh resolution is the same as that of the maximum level of refinement of the h-adaptive LDG method.