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.
A novel sampling method for reconstruction of a moving point acoustic source in $\mathbb{R}^3$
School of Mathematical Sciences, Sichuan Normal University（四川师范大学数学科学学院）；Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学与交叉科学研究院）
On the Connection between Residual Distribution Schemes and Flux Reconstruction
在这篇简短的论文中，我们考虑了\emph{Residual Distribution Schemes}（RD）和\emph{Flux Reconstruction}（FR）方法之间的联系。我们表明，通量重建可以被重铸到RD框架，反之亦然。由于这种密切的联系，我们能够应用已知的结果从RD计划FR方法。在这方面，我们提出了第一个示范的熵稳定性的FR计划正在考虑，并展示如何构建熵稳定的数值方案的基础上，我们的FR方法。同时，我们不将网格限制为张量结构或三角形元素，而是允许多边形。我们分析的关键是正确选择校正函数，我们在这里提出了一种方法。
A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method
Stability of Energy Stable Flux Reconstruction for the Diffusion Problem using the Interior Penalty and Bassi and Rebay II Numerical Fluxes for Linear Triangular Elements
Stable broken $H^1$ and $\bf H(\mathrm{div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Center for Data and Simulation Science (CDS), University of Cologne（科隆大学数据与模拟科学中心）
Minimization of an energy error functional to solve a Cauchy problem arising in plasma physics: the reconstruction of the magnetic flux in the vacuum surrounding the plasma in a Tokamak
Admissible Lax-Wendroff Flux Reconstruction Method with Automatic Differentiation on Adaptive Curved Meshes for Relativistic Hydrodynamics
Indian Institute of Technology Delhi（印度理工学院德里分校）；Johannes Gutenberg University Mainz（美因茨约翰内斯古腾堡大学）；IITD-Abu Dhabi Campus（德里印度理工学院阿布扎比分校）；Tata Institute of Fundamental Research（塔塔基础研究所）
$p$-robust equilibrated flux reconstruction in ${\boldsymbol H}(\mathrm{curl})$ based on local minimizations. Application to a posteriori analysis of the curl-curl problem
