标题： ADER-WENO 有限体积格式与时空自适应网格加密 显示英文标题

标题： ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement

摘要： 我们提出了第一个在多维空间中具有自适应网格加密（AMR）的高阶单步ADER-WENO有限体积格式。通过WENO重构获得高阶空间精度，而通过局部时空不连续Galerkin预测器方法实现高阶单步时间离散化。由于基础方案的一步性质，该算法特别适合在时空自适应网格上采用AMR策略，即具有时间精确的局部时间步进。AMR特性已通过“逐单元”实现，使用标准的树型算法，而该方案则通过消息传递接口（MPI）范式实现了并行化。新方案已在一系列非线性双曲守恒定律系统示例中进行了测试，包括经典的可压缩气体动力学欧拉方程和磁流体动力学（MHD）方程。通过数值收敛研究确认了空间和时间的高阶性，并还展示了相对于高度细化均匀网格的计算加速详细分析。我们还展示了所提出的高阶AMR方案明显优于传统二阶AMR方法的测试问题。首次将高阶ADER方法与二维和三维空间中的时空自适应网格相结合的该方案很可能成为计算物理、应用数学和力学等多个领域中有用的工具。 显示英文摘要

摘要： We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a standard tree-type algorithm, while the scheme has been parallelized via the Message Passing Interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space--time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.