High-order Space-time Flux Reconstruction Methods for Moving Domain Simulation

Abstract

A high-order space-time flux reconstruction (FR) method has been developed to solve conservation laws on moving domains. In the space-time framework, the moving domain simulation is similar to that on a stationary domain, except that the shape of the space-time elements varies with time (and space when a deforming grid is used). The geometric conservation law can be automatically satisfied to the level of the numerical resolution of the space-time schemes when the space-time discretization of the governing partial differential equations can resolve the geometric nonlinearity of curvilinear space-time elements. In this study, a space-time tensor product operation is used to construct the FR formulation, and the Gauss-Legendre quadrature points are used as solution points both in space and time. A dual time stepping method is used to solve the resulting space-time system. As has been proved by Huynh [J Sci Comput 96, 51 (2023)], in the temporal direction, the FR scheme with the Gauss-Legendre solution points is equivalent to the so-called DG-Gauss implicit Runge-Kutta (IRK) scheme when the quadrature rule based on the solution points (i.e. quadrature points used in DG) is sufficiently accurate to integrate the space-time curvilinear elements. Specifically, we show that when linear space-time elements are adopted in moving domain simulations, the temporal FR scheme based on Gauss-Legendre solution points can always guarantee its equivalency to IRK DG-Gauss. The conditions, under which the moving domain simulation with the method of lines are consistent with those using the space-time formulation, are also discussed. The new space-time FR method can achieve arbitrarily high-order spatial and temporal accuracy without numerical constraints on the physical time step in moving domain simulations. The temporal superconvergence property for moving domain simulations have been demonstrated.