In this article a new high order accurate cell-centered Arbitrary-Lagrangian-Eulerian (ALE) Godunov-type finite volume method with time-accurate local time stepping (LTS) is presented. The method is by construction locally and globally conservative. The scheme is based on a one-step predictor-corrector methodology in space-time and uses three main building blocks: First, a high order piecewise polynomial WENO reconstruction, to obtain a high order data representation in space from the known cell averages of the underlying finite volume scheme. Second, a high order space-time Galerkin predictor step based on a weak formulation of the governing PDE on moving control volumes. Third, a high order one-step finite volume scheme, based directly on the integral formulation of the conservation law in space-time. The algorithm being entirely based on space-time control volumes naturally allows for hanging nodes also in time, hence in this framework the implementation of a consistent and conservative time-accurate LTS becomes very natural and simple. The method is validated on some classical shock tube problems for the Euler equations of compressible gasdynamics and the magnetohydrodynamics equations (MHD). The performance of the new scheme is compared with a classical high order ALE finite volume scheme based on global time stepping. To the knowledge of the author, this is the first high order accurate Lagrangian finite volume method ever presented together with a conservative and time-accurate local time stepping feature.
Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows
Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation
High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids
University of Trento（特伦托大学）；Université de Pau et des Pays de l’Adour（波城及阿杜尔地区大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）
Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
A Posteriori Subcell Limiting of the Discontinuous Galerkin Finite Element Method for Hyperbolic Conservation Laws
University of Trento（特伦托大学）；CNRS and Institut de Mathématiques de Toulouse (IMT) Université Paul-Sabatier（法国国家科学研究中心与图卢兹数学研究所（IMT）保罗-萨巴蒂耶大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Two-Stage Fourth-order Gas-kinetic Scheme for Three-dimensional Euler and Navier-Stokes Solutions
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Hong Kong University of Science and Technology（香港科技大学）；HKUST Shenzhen Research institute（香港科技大学深圳研究院）
High-order discontinuous Galerkin schemes with subcell finite volume limiter and adaptive mesh refinement for a monolithic first-order BSSNOK formulation of the Einstein-Euler equations
