The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas.
Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space-time Galerkin formulation instead of the usual Cauchy-Kovalewski procedure.
The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions.
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
Efficient Implementation of ADER Schemes for Euler and Magnetohydrodynamical Flows on Structured Meshes -- Comparison with Runge-Kutta Methods
ADER（Arbitrary Derivative in Space and Time）方法用于求解双曲守恒律的时间演化问题，最近引起了人们的兴趣。ADER时间更新可以在单个步骤中执行，这在许多应用中是期望的。然而，以前的论文集中在理论上，而忽视了实现细节。本文的目的是使ADER计划访问提供两个有用的配方的方法，以及它们的实施细节的三维结构网格。因此，我们提供了一个详细的制定ADER计划的守恒律与非刚性源项在节点以及模态空间以及有用的实施相关的细节。我们还提供了详细的有效利用ADER计划在获得守恒律的数值通量以及电场的发散自由磁流体力学。本文还提出了一种基于WENO的从MHD的面心磁场获得区域平均磁场的有效方法。这里列出的方案已在第一作者的RIEMANN代码中实现。ADER计划的速度几乎是两倍的速度，强稳定性保持Runge-Kutta时间步进计划的精度，我们测试的所有订单。
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
Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
