一类混合MUSCL型E格式收敛性的研究

０．引 言 我们知道ＴＶＤ格式能高效地捕捉激波,特别是在激波附近解不发生振荡,因此在流体力学数值计算中得到了广泛的应用．但在 １９８５年,Ｊ．Ｇｏｏｄｍａｎ和 Ｒ．Ｌｅｖｅｑｕｅ证明了二维守恒型 ＴＶＤ格式至多是一阶精度格式［６］,而 ＭｍＢ格式［１］的提出使构造二维高精度格式有了新的突破．另外,在传统理论下,差分格式的熵条件是很难证明的．文［２］提出了在多维空间变量下的高精度差分格式下收敛性的一般方法．证明中只须假设格式解的一致ｌ∞估计和空间离散熵的弱估计,然后运用 Ｄｉｐｅｒｎａ的唯一性定理［５］…     

一种简化的三阶精度加权ENO格式

91．引言从七十年代后期开始，对双曲型守恒律方程数值方法的研究以VanLeer构造出来的MUSCL格式［‘］为先导，出现了一些全新的高分辨率守恒型差分格式．特别是A．Harten［‘］提出了TVD（TotalVariationDiminising）格式的概念后，双曲型守恒律方程数值方法的研究取得了飞速的发展．因为TVD格式可以保持数值解的单调性，所以它可以有效地抑制间断附近数值解的振荡，这方面有重要代表性的工作是［2－4］．由于TVD格式必须保证数值解的总变差不增，所以使得TVD格式在光滑解的局部极值点处降价．为了克服TVD格式的这个弱点，便出…     

求解多介质流体界面不稳定性问题的高精度WENO数值模拟方法

本文针对多介质流体界面不稳定性问题的数值模拟,把基于波传算法的高精度WENO数值格式用于守恒和非守恒形式的流体力学方程组计算。根据不同介质界面附近压强和速度保持一致的特点,求解了γ-model和体积分数形式的耦合型方程组,并与NND和NT2的模拟结果进行比较分析,表明该方法具有高分辨率和较强的捕捉界面的能力．     

正则长波方程的一个新的差分方法

§１．引言正则长波（ＲＬＷ）方程在１９６６年由Ｐｅｒｅｇｒｉｎｅ［１］第一次提出,它描述波的运动有与ＫｄＶ方程相同的逼近界,并且它能够相当好地摸拟ＫｄＶ方程的所有应用,因此引起了人们的注意．文［２－６］讨论了它的数值方法,其中文［２］提出了一个两层的和一个三层的差分格式,它们分别具有一阶和二阶精度．文［４］考虑了一个守恒的两层差分格式．本文考虑以下ＲＬＷ方程的初边值问题这个问题具有以下能量守恒律方程（３）的单个孤波解为其中ａ和是任意常数．从（７）可以明显地看出当一ｘＬ和ｘＲ足够大时,初边值问题（…     

对流占优扩散问题的并行计算

１．引言 在刻画流体运动的某些物理现象,以及研究热的传导、粒子的扩散等问题时,都会归结到求解对流扩散方程．用有限差分方法求解该方程,若采用显式方法,计算格式简单,但它们都是条件稳定的,时间步长必须取得非常小;若采用隐式方法,方法是无条件稳定的,但要解代数方程组,求解比较困难．Ｄ．Ｊ．ＥＶＡＮＳ和Ａ．Ｒ．ＡＨＭＡＤ在文［２］中提出了用显式交替方向法求解定态椭圆型方程,对Ｌａｐｌａｃｅ方程做了数值实验．本文将这个方法推广到了时间依赖的问题,而且适用于对流占优扩散问题的求解．基于二阶迎风格式［１］;本…     

一种基于WENO重构的半离散中心迎风格式

通过三阶WENO重构和半离散中心迎风数值通量的结合,给出了一种求解双曲型守恒律方程的三阶半离散中心迎风格式,格式保持了中心差分格式方法简单的优点．数值计算的结果表明该方法具有较高的分辨率．     

大气污染模型的POD基降维有限差分算法

通过特征正交分解法降低二维大气污染模型的经典差分格式的自由度,建立了一种降维差分迭代算法,分析了精确解与降维数值解之间最大模误差估计.对于相同的时间和空间步长,通过数值算例对比原始差分格式和降维格式的计算精度和时间,验证了降维算法的有效性和可行性.     

五阶精度WENO差分型格子波尔兹曼算法求解单守恒方程

给出了五阶精度WENO差分型格子波尔兹曼算法求解单守恒模型方程的计算方法.根据WENO差分格式的特点,定义了广义格子波尔兹曼分布函数,将守恒型方程的求解问题,转化成用WENO格式的差分算法对该分布函数进行求解.该方法的意义在于,将高精度高分辨率的WENO格式差分方法与近几十年发展起来的格子波尔兹曼方法相结合,从而很方便地构造出可以用于求解守恒型方程的格子波尔兹曼模型,使格子波尔兹曼方法在可压缩流领域的使用更简单.利用该方法分别构造了不同初值条件下的一维Burgers守恒型方程的求解模型,求出结果,并分析了模型的精度和稳定性.最后总结了方法的优点和不足,以及有待进一步研究解决的问题.     

关于非对称线性方程组的新迭代算法

５１．引言 二阶椭圆型非对称方程是一类重要的科学工程计算的数学模型,如对流扩散和油藏模拟方程等,有着广泛的实际应用背景．文献［２］和［３］基于原始微分方程及对应的离散问题提出了正定可对称化的新概念．基于这一概念及文山我们研究针对二维和三维二阶常系数非对称椭圆型方程数值模型的新选代算法,首先考虑下面的一维椭圆型问题：对区间［０,１］均匀剖分后得ｎ＋２节点,即ｘ;一ｉ·ｈ;ｉ＝０,··,,ｎ＋１,其中ｈ＝／（＋１）．如果用中心差分格式离散方程（１．１）,则在节点Ｘ;有如下差分方程： 一（１＋ｗＮ。;－…     

解Hamilton-Jacobi方程的无振荡局部加密方法

０．引 言 近年来,Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程（简称Ｈ－Ｊ方程）的数学理论与数值逼近已引起人们越来越多的关注．Ｈ－Ｊ方程不仅在原有的领域例如控制论、微分几何等有非常重要的应用［８］,而且不断开拓新的应用领域,例如用于网格生成［５］以及流体界面的水平集方法计算 ［９,１２,１３,１５］等．由于 Ｈ－Ｊ方程解的导数会出现间断,导致解曲面（线）出现尖点或纽结等现象［７］,故如何做到既节省计算时间,又能在光滑区域高精度数值求解和较好地分辨间断是一个十分重要的问题．文卜］通过在每个坐标方向构造单变量的…     

Numerical Methods for Euler Equations with Self-similar and Quasi Self-similar Solutions

Some problems of Euler equations have self-similar solutions which can be solved by more accurate method. The current paper proposes two new numerical methods for Euler equations with self-similar and quasi self-similar solutions respectively, which can use existing difference schemes for conservation laws and do not need to redesign specified schemes. Numerical experiments are implemented on one dimensional shock tube problems, two dimensional Riemann problems, shock reflection from a solid wedge, and shock refraction at a gaseous interface. For self-similar equations, one-dimensional results are almost equal to the exact solutions, and two-dimensional results also exhibit considerable high resolution. For quasi self-similar equations, the method can solve solutions that are not but close to self-similar, i.e. quasi self-similar, and this method can also achieve very high resolution when computing time is long enough. Numerical simulations to self-similar and quasi self-similar Euler equations have important implications on the study of self-similar problems, development of high resolution schemes, even the research for exact solutions of Euler equations.     

Kinetic flux vector splitting for radiation hydrodynamical equations

This paper is interested in the kinetic flux vector splitting (KFVS) for the multidimensional radiation hydrodynamical equations (RHEs) in zero diffusion limit. First, a generalized Maxwell–Boltzmann distribution function with two new parameters of temperature approximation is introduced to recover the macroscopic equations. These parameters are uniquely determined by macroscopic variables. Then, a high resolution KFVS method is proposed for the solution of the multidimensional RHEs. It does not require any Riemann solvers. Finally, several numerical examples are given to show the performance of our scheme.     

Modified kinetic flux vector splitting (m-KFVS) method for compressible flows

To resolve many flow features accurately, like accurate capture of suction peak in subsonic flows and crisp shocks in flows with discontinuities, to minimise the loss in stagnation pressure in isentropic flows or even flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order kinetic flux vector splitting (KFVS) method has been found to be very robust but suffers from the problem of having much more numerical diffusion than required, resulting in inaccurate computation of the above flow features. However, numerical dissipation can be reduced by refining the grid or by using higher order kinetic schemes. In flows with strong shock waves, the higher order schemes require limiters, which reduce the local order of accuracy to first order, resulting in degradation of flow features in many cases. Further, these schemes require more points in the stencil and hence consume more computational time and memory. In this paper, we present a low dissipative modified KFVS (m-KFVS) method which leads to improved splitting of inviscid fluxes. The m-KFVS method captures the above flow features more accurately compared to first order KFVS and the results are comparable to second order accurate KFVS method, by still using the first order stencil.     

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space-time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.     

Fast Euler solver for steady,one-dimensional flows

A numerical technique to solve the Euler equations for steady, one-dimensional flows is presented. The technique is essentially implicit, but is structured as a sequence of explicit solutions for each Riemann variable separately. Each solution is obtained by integrating in the direction prescribed by the propagation of the Riemann variables. The technique is second-order accurate. It requires very few steps for convergence, and each step requires a minimal number of operations. Therefore, it is three orders of magnitude more efficient than a standard time-dependent technique. The technique works well for transonic flows and provides shock fitting with errors as small as 0.001. Results are presented for subsonic and transonic problems. Errors are evaluated by comparison with exact solutions.     

Discontinuous Galerkin Methods Applied to Shock and Blast Problems

We describe procedures to model transient shock interaction problems using discontinuous Galerkin methods to solve the compressible Euler equations. The problems are motivated by blast flows surrounding cannons with perforated muzzle brakes. The goal is to predict shock strengths and blast over pressure. This application illustrates several computational difficulties. The software must handle complex geometries. The problems feature strong interacting shocks, with pressure ratios on the order of 1000 as well as weaker precursor shocks traveling rearward that also must be accurately captured. These aspects are addressed using anisotropic mesh adaptation. A shock detector is used to control the adaptation and limiting. We also describe procedures to track projectile motion in the flow by a level-set procedure.This revised version was published online in July 2005 with corrected volume and issue numbers.     

A numerical method for a kinetic equation and its application to propagating shock waves

An efficient numerical method for kinetic equations and its application to analyses of moving shock wave problems are presented. The present study aims to give an efficient scheme for two-dimensional unsteady gas flows. An explicit MacCormack difference method is applied to solve a BGK-model equation. The efficiency and accuracy of the scheme are examined in an application to one-dimensional shock structure problems. Furthermore, the scheme is applied to a two-dimensional flow problem: nonstationary reflection of a shock wave at a wedge. The present scheme is found to be useful and efficient for the analyses of two-dimensional unsteady rarefied gas flows.     

Numerical simulation of Richtmyer–Meshkov instability driven by imploding shocks

In this paper, the classical piecewise parabolic method (PPM) is generalized to compressible two-fluid flows, and is applied to simulate Richtmyer–Meshkov instability (RMI) induced by imploding shocks. We use the compressible Euler equations together with an advection equation for volume fraction of one fluid component as model system, which is valid for both pure fluid and two-component mixture. The Lagrangian-remapping version of PPM is employed to solve the governing equations with dimensional-splitting technique incorporated for multi-dimensional implementation, and the scheme proves to be non-oscillatory near material interfaces. We simulate RMI driven by imploding shocks, examining cases of single-mode and random-mode perturbations on the interfaces and comparing results of this instability in planar and cylindrical geometries. Effects of perturbation amplitude and shock strength are also studied.     

A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations

In this work we consider Runge–Kutta discontinuous Galerkin methods for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.