A high order central-upwind scheme for hyperbolic conservation laws

A high order central-upwind scheme for approximating hyperbolic conservation laws is proposed. This construction is based on the evaluation of the local propagation speeds of the discontinuities and Peer's fourth order non-oscillatory reconstruction. The presented scheme shares the simplicity of central schemes, namely no Riemann solvers are involved. Furthermore, it avoids alternating between two staggered grids, which is particularly a challenge for problems which involve complex geometries and boundary conditions. Numerical experiments demonstrate the high resolution and non-oscillatory properties of our scheme.     

双曲守恒律的几种新数值方法的比较研究

本文就一维线性双曲方程的光滑和间断两种初值问题的求解,对双曲守恒律的三种新数值方法,即,WENO方法、间断Galerkin方法和全局复合方法,进行了数值比较实验,在精度、计算速度等方面的比较上,对这三个方法有了一个较详细的了解,得到了一些有用的结论。     

Development and Comparison of Numerical Fluxes for LWDG Methods

The discontinuous Galerkin （DO） or local discontinuous Galerkin （LDG） method is a spatial discretization procedure for convection-diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. The Lax- Wendroff time discretization procedure is an altemative method for time discretization to the popular total variation diminishing （TVD） Runge-Kutta time discretizations. In this paper, we develop fluxes for the method of DG with Lax-Wendroff time discretization procedure （LWDG） based on different numerical fluxes for finite volume or finite difference schemes, including the first-order monotone fluxes such as the Lax-Friedfichs flux, Godunov flux, the Engquist-Osher flux etc. and the second-order TVD fluxes. We systematically investigate the performance of the LWDG methods based on these different numerical fluxes for convection terms with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.     

THE RELAXING SCHEMES FOR HAMILION-JACOBI EQUATIONS

1. IntroductionWe are illterested in the numerical approximation of viscosity solution of the following lirstorder Hamilton-Jacobi eqllationopt + H(&.,, rk..' ...) gb'.) - 0' (1'1)with initial data ac(~, 0) = ado(x). It is well known that the solutions to problem (1.1) typicallyare continuous (typically they are locally Lipschitz continuous) but with discoatinuous derivatives, even though the initial data ado E Coo. The nonuniqueness of such solutions to (1.1) alsonecessitates the introducti…     

溃坝问题的间断有限元方法

本文研究90年代初提出的Runge-Kutta间断Galerkin有限元方法，给出该方法的精度分析，通过经典算例验证该方法处理间断问题、捕捉锐利波形的能力，并将其推广到求解浅水问题．针对坝底无摩擦，无坡度的理想情形进行讨论，给出方溃坝和圆溃坝问题的数值模拟结果．     

基于坏单元指示子的p自适应RKDG算法

Runge-Kutta间断Galerkin(RKDG)方法是求解双曲守恒律方程的主流方法之一.很多双曲守恒律问题的规模特别巨大,数值计算求解时需要耗费大量的时间和计算机存储空间.为此我们设计了一个基于坏单元指示子的p自适应RKDG算法,它不仅能够保持原RKDG方法在光滑区域的高阶逼近,而且能够有效降低存储空间.同时,这也为后续进一步开展hp自适应RKDG算法的研究奠定了基础.     

四阶杆振动方程的两类高精度辛格式

In this paper, we present two classes of symplectic schemes with high order accuracy for solving four-order rod vibration equation utt uxxxx=0 via the third type generating function method. First, the equation of four order rod vibration is written into the canonical Hamilton system; second, overcoming successfully the essential difficult on the calculus of high order variations derivative, we get the semi-discretization with arbitrary order of accuracy in time direction for the PDEs by the third type generating function method. Furthermore the discretization of the related modified equation of original equation is obtained. Finally, arbitrary order accuracy symplectic schemes are obtained. Numerical results are also presented to show the effectiveness of the scheme, high order accuracy and properties of excellent long-time numerical behavior.     

HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS

In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes （HWENO-RK） of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.     

解高阶抛物型方程的高精度显式差分格式

1 引言 1960年,Saul’ev在文中讨论了如下的高阶(2m阶)抛物型方程 μ/t=(-1)~(m 1)~2mμ/x~(2m) (1)(其中m为正整数),提出了一类含极因子α的两层差分格式。当α=0时为显式格式,其稳定性条件为,r=△t/(△x)~(2m)<1/2~(2m-1),△t,△x分别为时间及空间步长。随后,文[2],[3]利用     

Systems of hyperbolic conservation laws with a resonant moving source

In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075-1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243-260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, L¹ stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515-1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094-1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu-Yang functional, one can prove the L¹ stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, L¹ stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226-1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075-1098] to a more general case.     

基于新光滑因子的WENO5格式

针对WENO格式的构造,本文给出了一个WENO为5阶的充分条件,降低了Henrick等人提出的充分条件对于权因子的精度要求.另外,对于Jiang和Shu提出的WENO5中的光滑因子中两项的系数做出了调整,并结合Borges等人的方法得到了新的WENO权因子计算方法.从数值试验的结果可以看出,新的WENO格式对于连续波形...     

An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In [35, 36], we presented an $h$-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws. A tree data structure (binary tree in one dimension and quadtree in two dimensions) is used to aid storage and neighbor finding. Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled "children". Extensive numerical tests indicate that the proposed $h$-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities. In this paper, we apply this $h$-adaptive method to solve Hamilton-Jacobi equations, with an objective of enhancing the resolution near the discontinuities of the solution derivatives. One- and two-dimensional numerical examples are shown to illustrate the capability of the method.     

解四阶抛物型方程高精度紧致差分格式

针对四阶抛物型方程周期初值问题,提出了一个两层隐式差分格式和一个三层隐式差分格式.它们的局部截断误差分别为O((Δt)2+(Δx)4)和O((Δt)2+(Δt)(Δx)2+(Δx)4),其中Δt,Δx分别为时间步长和空间步长.误差分析和数值实验均表明,本文构造的差分格式比经典的Crank-Nicolson格式和Saul’ev构造的差分格式精度更高.从精度及稳定性方面考虑,本文构造的格式也比文[5]的显式格式要好.     

On the hyperbolicity of certain models of polydisperse sedimentation

The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first‐order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Bürger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the models by Batchelor and Wen, Höfler and Schwarzer, and Davis and Gecol, for which the Jacobian of the flux is a rank‐3 or rank‐4 perturbation of a diagonal matrix. Explicit estimates of the regions of hyperbolicity of these models are derived via the approach of the so‐called secular equation (J. Anderson. Linear Algebra and Applications 1996; 246:49–70), which identifies the eigenvalues of the Jacobian with the poles of a particular rational function. Hyperbolicity of the system is guaranteed if the coefficients of this function have the same sign. Sufficient conditions for this condition to be satisfied are established for each of the models considered. Some numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.