A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

Abstract

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, det ∇ψ(x, t) = G○ψ(x, t). The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map ψ via composition of a sequence of near identity deformations of a uniform reference mesh. This allows us to utilize a simple, fast, and high order accurate implementation of the deformation method [21]. We design SAM to satisfy both internal and external consistency requirements between stability, accuracy, and efficiency constraints, and show that the scheme is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers [22,18], while running approximately 50-100 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2D gas dynamics. Specifically, we implement the C-method [66, 67] and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.