High-order Nodal Space-time Flux Reconstruction Methods for Hyperbolic Conservation Laws on Curvilinear Moving Grids

TL;DR

This paper introduces high-order nodal space-time flux reconstruction methods for solving hyperbolic conservation laws on curvilinear moving grids, emphasizing geometric conservation, aliasing error control, and superconvergence properties.

Contribution

It develops a space-time flux reconstruction framework that incorporates grid motion, analyzes aliasing errors, and demonstrates superconvergence in moving domain simulations.

Findings

The methods achieve temporal superconvergence of order (2k-1).

Aliasing errors can be effectively reduced with polynomial filtering.

Numerical experiments confirm robustness of superconvergence despite aliasing errors.

Abstract

High-order nodal space-time flux reconstruction (STFR) methods have been developed to solve hyperbolic conservation laws on curvilinear moving grids. Unlike the method-of-lines approach for moving domain simulation, the grid velocity is implicitly embedded into the curvilinear geometric representation of space-time elements. Several key issues in moving domain simulation, including the discrete geometric conservation law (GCL), solution and flux approximation, and aliasing error control, are discussed in the context of the nodal STFR framework. Conditions and the corresponding numerical strategies to reduce aliasing errors due to the curvilinear space-time representation of moving domain problems, including the discrete GCL errors (i.e. one type of aliasing errors in the space-time framework), are then explained and examined. Since a space-time tensor product is used to construct the FR formulation in this study, all space-time schemes show the temporal superconvergence property, similar to that presented by the implicit Runge-Kutta discontinuous Galerkin (IRK-DG) schemes, in moving domain simulation. Specifically, a nominal $k$th order scheme can achieve a ($2k-1$)th order superconvergence rate when solutions on $k$ Gauss-Legendre points are used to construct polynomials in the time dimension. The robustness of temporal superconvergence in the existence of aliasing errors induced by the curvilinear space-time representation, and upon de-aliasing operations based on polynomial filtering, has been examined with numerical experiments.