A High-Order Arbitrary Lagrangian-Eulerian Oscillation-Free Discontinuous Galerkin Scheme for One-Dimensional Compressible Multi-Material Flows (East Asian Journal on Applied Mathematics, 2025)
A high-order arbitrary Lagrangian-Eulerian (ALE) oscillation-free discontinuous Galerkin scheme for one-dimensional compressible multi-material flows is proposed.It couples a conservative equation in the volume-fraction model and the Euler equations of the fluid mixture dynamics. The mesh velocity is obtained by an adaptive meshmethod, which automatically concentrates mesh nodes near the regions with large gradients. This greatly reduce numerical dissipation at material interfaces. Besides, theresolution of the solution at material interfaces can be also improved. To control theoscillations, we add damping terms into the weak formulation of the system. Neitherparameter in new terms has to be artificially adjusted, and the difficulties with discontinuous solutions and complexities of designing limiters can be avoided. The scheme can beefficiently applied to compressible multi-material flows with essentially non-oscillatoryproperty and its steps are more concise than in indirect ALE methods. Although herewe only consider a third-order scheme, the method can be extended to any other orderby choosing suitable basis functions. Examples demonstrate the third-order accuracyand good performance of the scheme for one-dimensional compressible multi-materialflows.
Keywords: Compressible multi-material flow; arbitrary Lagrangian-Eulerian; oscillation-free DG scheme; adaptive mesh method; damping term.