双曲型守恒律的高阶、高分辨有限体积法

有限体积法是一种离散积分形式守恒律的数值方法。它可以吸 收有限元法和有限差分法的一些重要思想与技巧。由于它可方便地利 用多种类型的网格(结构网格和非结构网格)，从而非常适用于处理复 杂计算区域，目前已成为一种在计算流体力学中十分重要的方法。本 文将针对二维双曲守衡律，对高精度、高分辨的有限体积法及其近年 来的进展做一简要介绍。  

一类高精度TVD差分格式及其应用

构造了一维非线性双曲型守恒律的一个新的高精度、高分辨率的守恒型TvD差分格式。其构造思想是：首先，将计算区间划分为若干个互不相交的小区间，再根据精度要求等分小区间，通过各细小区间上的单元平均状态变量，重构各细小区间交界面上的状态变量，并加以校正；其次，利用近似Riemann解计算细小区间交界面上的数值通量，并结合高阶Runge—Kutta TVD方法进行时间离散，得到了高精度的全离散方法。证明了该格式的TVD特性。该格式适合于使用分量形式计算而无须进行局部特征分解。通过计算几个典型的问题，验证了格式具有高精度、高分辨率且计算简单的优点。  

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。  

双同守恒律方程的加权本质无振荡格式新进展

近几年,在计算流体力学中,高精度、高分辨率的加权本质无振荡(weighted essentially non-oscillatory , WENO)格式得到很大的发展.WENO格式的主要思想是通过低阶的数值流通量的凸组合重构得到高阶的逼近,并且在间断附近具有本质无振荡的性质.本文综合介绍了双曲守恒律方程的有限差分和有限体积迎风型WENO,中心WENO,紧致中心WENO以及优化的WENO格式等,讨论了负权的处理和多维问题的解决方法.最后,通过一些算例证明WENO格式的高精度,本质无振荡的性质.图6参40  

On an extension of the d’Alembert solution to initial–boundary value problems in multi-layered,multi-material domains

This work introduces a method for the exact solution of initial–boundary value problems for linear, one-dimensional conservation laws in multi-layered, multi-material domains. The method is based on the geometry of the solutions of such conservation laws and represents an extension of the d’Alembert solution to initial–boundary value problems in multi-layered, multi-material domains.  

Solitary wave solution to Aw-Rascle viscous model of traffic flow

A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries (KdV) equation. A numerical simulation is conducted by the first-order accurate Lax-Friedrichs scheme, which is known for its ability to capture the entropy solution to hyperbolic conservation laws. Periodic boundary conditions are applied to simulate a lengthy propagation, where the profile of the derived KdV solution is taken as the initial condition to observe the change of the profile. The simulation shows good agreement between the approximated KdV solution and the numerical solution.  

Improved third‐order weighted essentially nonoscillatory scheme

A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.  

A novel computational method for modelling stochastic advection in heterogeneous media

The paper is devoted to a new computational method for problems of transport in highly non-uniform media. In particular, the method is applied to the problem of anomalous contaminant transport in a field with a randomly distributed permeability, which was modelled as a stochastic advection process governed by a stochastic advection model. The stochastic advection model is used to generate different realisations of micro-dispersion parameters required for direct numerical simulations. The new numerical method combines the merits of finite-volume and finite-difference approaches and is demonstrated to be efficient and robust in several benchmark advection tests. For the stochastic advection problem considered the results of the new computational method are in a good agreement with analytical predictions available for different stochastic advection regimes.  

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tⁿ is connected to the new one at t^n + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.  

Anisotropic slope limiting for discontinuous Galerkin methods

In this paper, we present an anisotropic version of a vertex‐based slope limiter for discontinuous Galerkin methods. The limiting procedure is carried out locally on each mesh element utilizing the bounds defined at each vertex by the largest and smallest mean value from all elements containing the vertex. The application of this slope limiter guarantees the preservation of monotonicity. Unnecessary limiting of smooth directional derivatives is prevented by constraining the x and y components of the gradient separately. As an inexpensive alternative to optimization‐based methods based on solving small linear programming problems, we propose a simple operator splitting technique for calculating the correction factors for the x and y derivatives. We also provide the necessary generalizations for using the anisotropic limiting strategy in an arbitrary rotated frame of reference and in the vicinity of exterior boundaries with no Dirichlet information. The limiting procedure can be extended to elements of arbitrary polygonal shape and three dimensions in a straightforward fashion. The performance of the new anisotropic slope limiter is illustrated by two‐dimensional numerical examples that employ piecewise linear discontinuous Galerkin approximations. Copyright © 2017 John Wiley & Sons, Ltd.