Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory(WENO)schemes to solve the one-dimensional and two-dimensional shallow water equations with source terms.Multilevel time discretization methods can make full use of computed information by WENO spatial discretization and save CPU cost by holding the former computational values.Extensive simulations are performed,which indicate that,the finite difference WENO schemes with multilevel time discretization can achieve higher accuracy,and are more cost effective than WENO scheme with Runge-Kutta time discretization,while still maintaining nonoscillatory properties.

双曲型守恒律的一类局部化的高效差分格式

构造了一维非线性双曲型守恒律的一类局部化的高效全离散差分格式,并将该格式推广到一维守恒方程组及二维守恒方程(组).最后,给出了几个标准算例.数值计算结果表明此格式具有高精度高分辨激波、稀疏波和接触间断,且边界条件易于处理等优点.

一个三阶基本无振荡的差分格式

利用待定系数法推导了一阶空间变量导数的三阶差分逼近,进而将通常对流通量的逼近方法推广到对通量导数的逼近.通过引入TVD限制器函数,对导数的三阶差分逼近进行校正,并结合三阶Runge-Kutta时间离散方法,构造了求解双曲型守恒律方程的一个三阶基本无振荡差分格式.该格式具有形式简单、计算量小且易于推广到非均匀网格等优点.通过给出的两个标准数值算例,验证了格式的有效性.

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

As the generalization of the integer order partial differential equations(PDE),the fractional order PDEs are drawing more and more attention for their applications in fluid flow,finance and other areas.This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin(DG)methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in space.The Caputo derivative is chosen as the representation of spatial derivative,because it may represent the fractional derivative by an integral operator.Some numerical examples show that the convergence orders of the proposed local Pk–DG methods are O(hk+1)both in one and two dimensions,where Pk denotes the space of the real-valued polynomials with degree at most k.

结构网格FVWENO格式构造及浸入边界Ghost cell方法研究

首先构造二维FVWENO(finite volume weighted essentially non-oscillatory)格式:在二维空间模板上构造一个3次多项式.将此模板分割为8个较小的模板,每一个小模板上包含6个单元.在每一个小模板上构造一个2次多项式,要求多项式在单元上的平均值与需重构的物理量在同一单元上的平均值相等.之后,通过计算得到线性权,光滑指示器,非线性权.然后利用TVD Runge-Kutta时间离散方法得到时空一致高阶精度格式.最后,结合两种浸入边界Ghost cell方法,能针对复杂物体流动问题在结构网格上进行正确的数值模拟.

A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes

In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws.The key idea of HWENO is to evolve both with the solution and its derivative,which allows for using Hermite interpolation in the reconstruction phase,resulting in a more compact stencil at the expense of the additional work.The main difference between this work and the formal one is the procedure to reconstruct the derivative terms.Comparing with the original HWENO schemes of Qiu and Shu,one major advantage of new HWENOschemes is its robust in computation of problem with strong shocks.Extensive numerical experiments are performed to illustrate the capability of the method.

虚拟单元有限体积WENO5格式及其应用

本文在笛卡尔网格上给出一种五阶有限体积加权基本无振荡格式首先在二十五个单元构成的空间大模板上构造五次不完全多项式;将此大模板划分为九个子模板,并在其上构造三次不完全多项式;计算线性权,光滑指示器和非线性权;利用三阶TVD Runge-Kutta时间离散方法得到时空一致高精度格式.虽然该格式具有较高数值精度但不能直接应用于具有复杂拓扑结构物体的可压缩绕流问题.为降低该格式对网格的要求,本文采用ST和GBCM两种浸入边界虚拟单元方法处理物面边界条件,将有限体积高精度格式同虚拟单元方法相结合,能有效降低格式构造和网格生成的复杂性.文中给出的多个经典复杂物体绕流问题的数值计算充分表明了本方法的可靠性和有效性.