A Well-Balanced HLL Scheme for Hyperbolic Conservation Systems With Source Terms北大核心CSCD

针对含源项的双曲守恒方程给出了一种新的有限体积格式.经典的有限体积格式不能正确地模拟对流通量项和外力之间的平衡所产生的动力学问题.为解决这个问题,仿照经典的HLL近似Riemann求解器设计思路设计了含源项的近似Riemann求解器.针对含重力源项的一维流体Euler方程和理想磁流体方程,通过对通量计算格式的修正得到了保平衡HLL格式（WB-HLL）,并给出了保平衡的证明.针对一维Euler方程和理想磁流体给出了两个算例,比较了传统HLL格式和提出的WB-HLL格式的计算精度.计算结果表明,WB-HLL格式精度更高,收敛更快.     

A Finite Volume Scheme for Savage-Hutter Equations on Unstructured Grids

A Godunov-type finite volume scheme on unstructured grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate. We show the direct observation that the model isn't a Galilean invariant system. At the cell boundary, the modified Harten-Lax-van Leer (HLL) approximate Riemann solver is adopted to calculate the numerical flux. The modified HLL flux is not troubled by the lack of Galilean invariance of the model and it is helpful to handle discontinuities at free interface. Rigidly the system is not always a hyperbolic system due to the dependence of flux on the velocity gradient. Even so, our numerical results still show quite good agreements with reference solutions. The simulations for granular avalanche flows with shock waves indicate that the scheme is applicable.     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.     

Numerical solutions of diffusion mathematical models with delay

In this paper an explicit numerical difference scheme for mixed problems for the delay diffusion equation is proposed, as a generalization of the classic difference scheme for the diffusion problem. A sufficient condition for the asymptotic stability of the new scheme is proved. Consistence, convergence and some properties of stability for this scheme are studied. Illustrative examples of numerical results are also included.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

High‐order asymptotic‐preserving schemes for linear systems: Application to the Goldstein–Taylor equations

In this work, we address the numerical approximation of linear systems with possibly stiff source terms which induce an asymptotic diffusion limit. More precisely, we are interested in the design of high‐order asymptotic‐preserving schemes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme. This alteration can be understood as a numerical diffusion reduction technique and allows to capture the correct asymptotic behavior in the diffusion limit and to consider uniformly high‐order extensions. We more specifically consider the case of the Goldstein–Taylor model but the overall approach is shown to be easily adapted to more general systems.     

An Efficient Method for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations

We propose a simple numerical method for calculating both unsteady and steady state solution of hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography,and the quasi one-dimensional nozzle flows. We use the interface value, rather than the cell-averages, for the source terms, which results in a well-balanced scheme that can capture the steady state solution with a remarkable accuracy. This method approximates the source terms via the numerical fluxes produced by an (approximate) Riemann solver for the homogeneous hyperbolic systems with slight additional computation complexity using Newton‘s iterations and numerical integrations. This method solves well the subor super-critical flows, and with a transonic fix, also handles well the transonic flows over the concentration. Numerical examples provide strong evidence on the effectiveness of this new method for both unsteady and steady state calculations.     

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and then we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.     

关于广义KPP方程的数值解

广义KPP(Kolmogorov-Petrovskii-Piskunov)方程是一个积分微分方程.为了要研究其数值解,我们首先将该方程转化为一个非线性双曲型方程,然后构造了一个线性化的差分格式,得到了差分格式解的存在唯一性,利用能量不等式证明了差分格式二阶收敛性和关于初值的无条件稳定性,数值结果验证了本文提出的方法.     

A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment method

In this article, we consider a multi‐species kinetic model which leads to the Maxwell–Stefan equations under a standard diffusive scaling (small Knudsen and Mach numbers). We propose a suitable numerical scheme which approximates both the solution of the kinetic model in rarefied regime and the one in the diffusion limit. We prove some a priori estimates (mass conservation and nonnegativity) and well‐posedness of the discrete problem. We also present numerical examples where we observe the asymptotic‐preserving behavior of the scheme.     

二相流计算的一种差分算法

考虑两相流的力学行为,忽略相间的耗散作用,建立了Euler型的基本控制方程．状态方程采用刚性状态方程．基于Abgrall提出的准则,在流动区域内,对可压两相流提出了一个精度较高的Euler型数值方法,数值格式是Godunov型格式,对守恒型和非守恒型方程采用HLLC型和Lax-Friedrichs型近似Riemann解算器,引入了速度驰豫和压强驰豫过程来代替两相间的相互作用．在一维情形下给出数值算例,并且和Saurel的算例进行了比较,结果表明该算法不但精确而且稳定,且在间断处没有数值振荡．     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

三维多面体网格上扩散方程的保正格式

 针对三维任意(星形)多面体网格, 本文构造了扩散方程的一种单元中心型非线性有限体积格式, 证明了该格式具有保正性. 在该格式设计中, 除引入网格中心量外, 还引入网格节点量和网格面中心量作为中间未知量, 它们将用网格中心未知量线性组合表示, 使得格式仅有网格中心未知量作为基本未知量. 在节点量计算中, 利用网格面上的调和平均点, 设计了一种适用于三维多面体网格的局部显式加权方法. 该格式适用于求解非平面的网格表面和间断扩散系数的问题. 数值例子验证了它对光滑解具有二阶精度和保正性.     

Implicit subgrid‐scale modeling by adaptive local deconvolution

A class of implicit Subgrid‐Scale (SGS) models for Large‐Eddy Simulation (LES) is obtained from a new approach for the finite‐volume discretization of hyperbolic conservation laws. The extension of a standard deconvolution operator and the choice of a suitable numerical flux function result in a truncation error which can be forced to have physical significance. As determined by a modified‐differential‐equation analysis, the free parameters of this implicit SGS model can be adjusted to approximate a known explicit SGS model. Computational results for the viscous Burgers equation show that the model with parameters identified by evolutionary optimization gives significantly better predictions than other models. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic preserving scheme for hydrodynamics radiative transfer models

In this paper, we present a numerical scheme for a hydrodynamics radiative transfer model consisting of two steps: the first one is based on a relaxation method and the second one on the well balanced scheme. The derivation of the scheme relies on the resolution of a stationary Riemann problem with source terms. The obtained scheme preserves the limited flux property and it is compatible with the diffusive regime of hydrodynamics radiative transfer models. These properties are illustrated by numerical tests, one of them involves a radiative transfer model coupled with an equation for the temperature of the material.     

Low-Frequency Multidimensional Instabilities for Reacting Shock Waves

This paper presents a weakly nonlinear analysis for one scenario for the development of transversal instabilities in detonation waves in two space dimensions. The theory proposed and developed here is most appropriate for understanding the behavior of regular and chaotically irregular pulsation instabilities that occur in detonation fronts in condensed phases and occasionally in gases. The theory involves low-frequency instabilities and through suitable asymptotics yields a complex Ginzburg-Landau equation that describes simultaneously the evolution of the detonation front and the nonlinear interactions behind this front. The asymptotic theory mimics the familiar theory of nonlinear hydrodynamic instability in outline; however, there are several novel technical aspects in the derivation because the phenomena studied here involve a complex free boundary problem for a system of nonlinear hyperbolic equations with source terms.     

求解对流扩散方程的一致四阶紧致格式

目前,许多高精度差分格式,由于未成功地构造与其精度匹配的稳定的边界格式,不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式,它的边界点的计算格式和内点的计算格式的截断误差主项保持一致,给出了具体内点和边界格式;并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度;在对流占优情况下,本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小,取得了不错的效果,理论结果得到了数值验证;驱动方腔数值结果显示,本文对N-S方程的离散格式具有很好的可靠性,适合对复杂流体流动的数值模拟和研究.     

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semi-linear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. The convergence of the approximate solution toward a weak solution is established in the cases of first and second order accurate MUSCL relaxed methods.

     

Asymptotic‐preserving Godunov‐type numerical schemes for hyperbolic systems with stiff and nonstiff relaxation terms

We devise a new class of asymptotic‐preserving Godunov‐type numerical schemes for hyperbolic systems with stiff and nonstiff relaxation source terms governed by a relaxation time ε. As an alternative to classical operator‐splitting techniques, the objectives of these schemes are twofold: first, to give accurate numerical solutions for large, small, and in‐between values of ε and second, to make optional the choice of the numerical scheme in the asymptotic regime ε tends to zero. The latter property may be of particular interest to make easier and more efficient the coupling at a fixed spatial interface of two models involving very different values of ε. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A numerical method for solving the hyperbolic telegraph equation

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008