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

TL;DR

This paper develops high-order nodal space-time flux reconstruction (STFR) methods for hyperbolic conservation laws on curvilinear moving grids, embedding grid motion within space-time element geometry and addressing discrete GCL, flux/solution accuracy, and aliasing control. It establishes a rigorous mathematical foundation for tensor-product STFR on general space-time grids, analyzes the interplay between GCL and aliasing, and introduces a hybrid reference-physics implementation plus projection-based polynomial filtering to manage aliasing and preserve freestream. A key finding is the temporal superconvergence of order $2k-1$ in time for a nominal spatial order of $k$, maintained even on moving grids when geometry is accurately represented; geometric nonlinearity can influence this rate, necessitating finer resolution or filtering. The work provides practical strategies to realize high-order accuracy on moving domains, with implications for 4D space-time simulations and moving-boundary Navier–Stokes problems, and suggests directions for linking space-time methods with traditional method-of-lines approaches.

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.

Figures (16)

• Figure 1: High-order curvilinear (a) 2D and (b) 3D tensor-product space-time elements. For the 2D space-time element in (a), the $P^1$ polynomial is used to construct the 1D spatial element, and the $P^2$ polynomial is used to construct the curve along the time dimension. For the 3D space-time element in (b), $P^2$ polynomials are used to construct the spatial and temporal curves along each dimension. Note that the edge, surface, and volume points for the 3D space-time element are not numbered in (b).
• Figure 2: Distribution of solution points (circle) and flux points (square) in (a) 2D and (b) 3D tensor-product space-time elements. Gauss-Legendre points are used in each dimension to construct solution and flux polynomials. For the 2D space-time element in (a), the $P^3$ polynomial is used to construct the solution along the spatial dimension, and the $P^2$ polynomial is used to construct the solution along the time dimension. For the 3D space-time element in (b), $P^2$ polynomials are used to construct the solution along each (i.e., $\xi$- and $\eta$-) spatial dimension, and the $P^1$ polynomial is used to construct the solution along the time dimension.
• Figure 3: (a) Spatial convergence rates from $P^1$ to $P^5$ spatial constructions; and (b) temporal convergence rates from $P^1$ to $P^4$ temporal constructions for the 1D wave propagation problem on a stationary grid at $t=1$ (i.e. one wave propagation period).
• Figure 4: (a) Spatial convergence rates from $\mathbb{Q}^2$ to $\mathbb{Q}^5$ spatial constructions for the 2D Euler vortex propagation problem on a stationary grid at the 1/8 wave propagation period; and (b) temporal convergence rates from the $P^1$ to $P^3$ temporal constructions evaluated at one wave propagation period.
• Figure 5: (a) The wave field and the corresponding deformed grid at $t=t_{max}=0.2$ for the 2D linear wave equation. (b) The grid with $16 \times 16$ bilinear elements plotted on top of that with $8 \times 8$ bilinear elements.

Theorems & Definitions (12)

• Lemma 1
• proof
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Theorem 2
• proof
• Theorem 3
• proof
• Remark 4.4