Curvilinear grids for WENO methods in astrophysical simulations

We investigate the applicability of curvilinear grids in the context of astrophysical simulations and WENO schemes. With the non-smooth mapping functions from Calhoun et al. (2008), we can tackle many astrophysical problems which were out of scope with the standard grids in numerical astrophysics. We describe the difficulties occurring when implementing curvilinear coordinates into our WENO code, and how we overcome them. We illustrate the theoretical results with numerical data. The WENO finite difference scheme works only for high Mach number flows and smooth mapping functions, whereas the finite volume scheme gives accurate results even for low Mach number flows and on non-smooth grids.     

封闭空间爆炸载荷特性研究

封闭空间爆炸载荷主要包含瞬态冲击波和持续时间较长的准静态超压。为了研究封闭空间爆炸载荷特性,基于FORTRAN平台,采用三阶WENO有限差分格式编写了爆炸波高精度三维数值计算程序。应用Sod激波管、双爆轰波碰撞等经典算例验证了所开发数值程序的可靠性。在封闭空间内炸药爆炸波数值计算的基础上,基于冲量等效原则提出封闭空间内爆炸载荷简化模型,理论推导给出准静态超压峰值计算公式并通过数值计算结果验证了该公式的可靠性。开发的高精度爆炸波三维数值计算程序及提出的简化载荷模型可用于封闭空间内爆炸载荷的快速计算,为工程抗爆结构设计提供载荷输入。     

Numerical Comparison of WENO Finite Volume and Runge–Kutta Discontinuous Galerkin Methods

High order WENO (weighted essentially non-oscillatory) schemes and discontinuous Galerkin methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In this paper we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing.     

Truncation error,dissipation and dispersion terms of fifth order WENO and of WCS for 1D conservation law

In this paper, a derivation and a comparison of the truncation errors and the dissipation and dispersion terms of the fifth-order weighted essentially non-oscillatory scheme and of the weighted compact scheme are presented. The schemes are compared for smooth functions (by Fourier analysis), and near a shock.     

A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws II: Systems of Equations

A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in [36]. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the real state, and one corresponding to a ghost node state, analogous to the Ghost Fluid Method of [12]. High order pointwise convergence was demonstrated for scalar linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions in the first paper of this series [3]. The solutions here are compared to standard high order shock capturing schemes, when appropriate. This paper focuses on the issues involved in tracking discontinuities in systems of conservation laws. Examples will be presented of tracking contacts and hydrodynamic shocks in inert and chemically reacting compressible flow.     

Physis语言框架在WENO高阶数值格式异构计算中的应用

WENO(weighted essentially non-oscillatory)是计算流体力学中广泛采用的一种高阶数值格式。由于算法本身和异构计算编程的复杂性,需要开展异构计算代码自动生成的研究,以加速更多的应用。本文基于Physis这一领域编程语言框架,针对三维五阶WENO计算的天文应用,实现了其异构代码的自动生成。在超级计算机"元"上的测试结果表明,自动生成的异构计算代码具有良好的可扩展性,计算性能达到手工优化异构代码的72%,可为相关流体计算的异构代码生成提供借鉴。     

二维浅水方程的高阶松弛格式求解

利用松弛方法,将二维浅水方程转化为松弛方程组,并用逐维五阶WENO重构和显隐式Runge-Kutta方法对松弛方程组的空间和时间方向进行离散,建立了求解二维浅水方程的五阶松弛格式。WENO重构方法的引入既提高了格式的精度,又可保证格式是无振荡的。应用该格式对圆柱溃坝等问题进行了数值模拟,计算结果与用其它方法所得结果吻合,表明了方法的有效性。     

基于WENO格式的二维水流水质数值模拟

应用高精度WENO(Weighted Essentially Non-oscillatory Schemes)格式结合有限体积法建立了二维水流水质的数学模型.在求解过程中,将单元体上的面积分化为单元边界上的线积分,将二维问题化为一维问题.采用FDS(Flux Difference Splitting)格式求解跨单元边界的法向通量.采用此模型对连续弯道水流和污染物的扩散输移进行了数值模拟,得到了流场及岸边污染物排放的浓度场,并与物理试验结果进行了比较,表明这类格式精度高,稳定性能好.该模型可为工程领域中的水流水质问题的处理提供依据,具有较大的实用价值.     

求解一维双曲守恒律方程的网格自适应细化算法

采用基于一类半离散Godunov型中心迎风算法的网格自适应细化算法求解一维双曲守恒律方程.在时间方向上采用三阶TVD-Runge-Kutta格式,空间离散采用WENO方法.同时借助一个新的间断探测器来判断间断区域,并对这个区间进行网格加密,使其数值解的精度更高.在数值算例中,通过对非网格自适应和网格自适应结果的比较,可以得出,网格自适应细化算法可以很好的捕捉间断区域,且能有效地抑制非物理震荡的产生.     

A Numerical Study on Large-Time Asymptotics of the Lifshitz–Slyozov System

We numerically investigate the long time behavior of solutions of the Lifshitz–Slyozov system. We propose a numerical scheme in which the numerical dissipation is controlled in such a way that the results for large time are meaningful. In this respect, we find the long time behavior to crucially depend on the distribution of largest aggregates present in the solution. This fact proved, in some particular cases in (23), was difficult to obtain with previous numerical schemes in the engineering literature leading to wrong statements. We propose a numerical scheme in which we can observe and quantify the equilibration rates towards the right asymptotic profile. Moreover, this system appears to be a very interesting test problem for any anti dissipative scheme for conservation laws.