This work introduces a single-stage, single-step method for the compressible Euler equations that is provably positivity-preserving and can be applied on both Cartesian and unstructured meshes. This method is the first case of a single-stage, single-step method that is simultaneously high-order, positivity-preserving, and operates on unstructured meshes. Time-stepping is accomplished via the Lax-Wendroff approach, which is also sometimes called the Cauchy-Kovalevskaya procedure, where temporal derivatives in a Taylor series in time are exchanged for spatial derivatives. The Lax-Wendroff discontinuous Galerkin (LxW-DG) method developed in this work is formulated so that it looks like a forward Euler update but with a high-order time-extrapolated flux. In particular, the numerical flux used in this work is a linear combination of a low-order positivity-preserving contribution and a high-order component that can be damped to enforce positivity of the cell averages for the density and pressure for each time step. In addition to this flux limiter, a moment limiter is applied that forces positivity of the solution at finitely many quadrature points within each cell. The combination of the flux limiter and the moment limiter guarantees positivity of the cell averages from one time-step to the next. Finally, a simple shock capturing limiter that uses the same basic technology as the moment limiter is introduced in order to obtain non-oscillatory results. The resulting scheme can be extended to arbitrary order without increasing the size of the effective stencil. We present numerical results in one and two space dimensions that demonstrate the robustness of the proposed scheme.
A combined Lax-Wendroff/interpolation approach with finite element method for a three-dimensional system of tectonic deformation model: application to landslides in Cameroon
A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers. I. Hyperbolic Conservation Laws
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（计算物理实验室，应用物理与计算数学研究所）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
An L-Stable Sequential Implicit Two-Stage Fourth-Order Time Discretization with Second-Order Stiff Damping
School of Mathematics and Information Science, Henan Polytechnic University（河南理工大学数学与信息科学学院）；School of Mechatronical Engineering, Beijing Institute of Technology（北京理工大学机械与车辆学院）
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）
Weighted Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Methods for Conservation Laws
University of Science and Technology of China（中国科学技术大学）；Laoshan Laboratory（崂山实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Quantifying Resolutions for DNS and LES for Lax-Wendroff Method: Application to Uniform/Non-Uniform Compact Schemes
数值方法的全局谱分析（GSA）确保了色散关系保持（DRP）属性除了确保数值稳定性之外还被校准，如冯·诺依曼分析中所提倡的。DRP性质在必须保留控制方程和边界条件的时空依赖性的情况下起主要作用，例如在流体流动过渡的直接数值模拟（DNS）和大涡模拟（LES）中。使用GSA的概念，基于用于时间积分的Lax-Wendroff方法的方法使用在"Hybrid sixth order spatial discretization scheme for non-uniform Cartesian grids-Sharma et al. Comput.流体，157，208 - 231（2017）。"用于此分析的模型方程是一维（1D）对流扩散方程（CDE），它为Lax-Wendroff方法提供了一个独特的状态，其结果将对纳维-斯托克斯方程的求解产生直接影响。此外，控制方程的具体选择使得能够直接评估用于求解流体流动的数值方法的性能，这是由于其与纳维-斯托克斯方程一一对应，如在"Effects of numerical anti-diffusion in closed nonstable flows dominated by two-dimensional纳维-斯托克斯方程-Suman et al. Comput.流体，201，104479（2020）"。考虑了非均匀紧致格式的极限情况，即均匀网格。用GS…
A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
University of Wisconsin–Platteville（威斯康星大学普拉特维尔分校）；Instituto Tecnológico Autónomo de México（墨西哥自治理工学院）；Iowa State University（爱荷华州立大学）；Minnesota State University Moorhead（明尼苏达摩尔黑德州立大学）；University of Florida（佛罗里达大学）
Determining Parameter Ranges for High Accuracy Large Eddy Simulation by Lax-Wendroff Method
采用傅里叶变换，在谱平面上用一般的修正微分方程（MDE）方法对Lax-Wendroff（LW）方法进行了分析。在这种方法中，色散关系的概念起着重要的作用有关的空间和时间的依赖性的控制微分方程，包括初始和边界条件，在开发高精度的计划。这种色散关系保持计划校准在频谱平面使用的整体谱分析的数值方法在全域。在这个框架内，数值方法进行校准，通过研究对流和扩散的基本物理过程，这个典型的模型问题。在LW方法中，空间和时间的离散化被认为是在一起，与时间的衍生物取代相应的空间衍生物使用的控制方程。在这里的LW方法研究对流扩散方程（CDE）建立一个显式的中心差分格式，调用第三和第四空间导数的MDE，在其一般形式的数值参数的限制。因此，对于LW方法，得到两个不同的MDE，这取决于LW方法是仅应用于对流算子，还是同时应用于对流和扩散算子。受纳维-斯托克斯方程与“Effects of numerical anti-diffusion in closed nonstable flows dominated by two-dimensional纳维-斯托克斯方程- Suman et al. Fluids，201，104479（2020）"，本文对LW方法的这两种变体解决流动问题进行了评估。除了映射的数值特性进行大涡模拟的LW方法，模拟的规范盖驱动空腔问题的超临界雷诺数的均匀网格。
The Picard integral formulation of weighted essentially non-oscillatory schemes
Department of Mathematics, Michigan State University（密歇根州立大学数学系）；Department of Electrical and Computer Engineering, Michigan State University（密歇根州立大学电气与计算机工程系）
Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections
A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density
Chongqing University（重庆大学）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；School of arts and sciences, Chengdu College of University of Electronic Science and Technology of China（电子科技大学成都学院文理学院）；Sichuan International Studies University（四川外国语大学）
