Arbitrary Lagrangian-Eulerian Methods for Compressible Flows

In this report, we propose a collection of methods to make such an approach possible for Euler equations in one and two dimensions. We propose an explicit single-step ALE DG scheme for hyperbolic conservation laws. The scheme considerably reduces the numerical dissipations introduced by the Riemann solvers. We show that the scheme also preserves the constant states for any mesh motion. We then study the effect of mesh quality on the accuracy of the simulations, and based on that, come up with a mesh quality indicator for the ALE DG method. Based on the considerations from the study on mesh quality, we design a local mesh velocity algorithm to compute the motion of the mesh. And finally, we propose a local mesh adaptation algorithm to control the quality of the mesh, and prevent the mesh from degradation.