二维抛物型方程的高精度多重网格解法

提出了数值求解二维抛物型方程的一种新的高精度加权平均紧隐格式，利用Fourier分析方法证明了该格式是无条件稳定的，为了克服传统迭代法在求解隐格式是收敛速度慢的缺陷，利用了多重网格加速技术，大大加快了迭代收敛速度，提高了求解效率，数值实验结果验证了方法的精确性和可靠性。     

时间分数阶期权定价模型的一类有效差分方法

时间分数阶期权定价模型(时间分数阶Black-Scholes方程)数值解法的研究具有重要的理论意义和实际应用价值.对时间分数阶Black-Scholes方程构造了显-隐格式和隐-显差分格式,讨论了两类格式解的存在唯一性,稳定性和收敛性.理论分析证实,显-隐格式和隐-显格式均为无条件稳定和收敛的,两种格式具有相同的计算量.数值试验表明:显-隐和隐-显格式的计算精度与经典Crank-Nicolson(C-N)格式的计算精度相当,其计算效率(计算时间)比C-N格式提高30%.数值试验验证了理论分析,表明本文的显-隐和隐-显差分方法对求解时间分数阶期权定价模型是高效的,证实了时间分数阶Black-Scholes方程更符合实际金融市场.     

求解一维定常对流扩散反应方程的混合型紧致差分格式

借助显式紧致格式和隐式紧致格式的思想,基于截断误差余项修正,并结合原方程本身,构造出了一种求解一维定常对流扩散反应方程的高精度混合型紧致差分格式.格式仅用到三个点上的未知函数值及一阶导数值,而一阶导数值利用四阶Pade格式进行计算,格式整体具有四阶精度.数值实验结果验证了格式的精确性和可靠性.     

一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性 Schrödinger 方程给出两个紧致差分格式, 运用能量方法和两个新的分析技 巧证明格式关于离散质量和离散能量守恒, 而且在最大模意义下无条件收敛. 对非线性紧格式构造了 一个新的迭代算法, 证明了算法的收敛性, 并在此基础上给出一个新的线性化紧格式. 数值算例验证 了理论分析的正确性, 并通过外推进一步提高了数值解的精度.     

时间分数阶扩散方程的数值解法

分数阶微分方程在许多应用科学上比整数阶微分方程更能准确地模拟自然现象.考虑时间分数阶扩散方程,将一阶的时间导数用分数阶导数α(0<α<1)替换,给出了一种计算有效的隐式差分格式,并证明了这个隐式差分格式是无条件稳定和无条件收敛的,最后用数值例子说明差分格式是有效的.     

三维非线性Schr?dinger方程的六阶精度紧致ADI分裂方法

提出了一种结合二阶Strang分裂技术的六阶紧致交替方向隐式方法,用于求解三维非线性Schr?dinger方程.方法在时间上具有二阶精度,在空间上具有六阶精度.稳定性分析表明,方法是无条件稳定的.通过数值实验验证了方法满足守恒律,并为三维非线性Schr?dinger方程提供了精确、稳定的解.     

时间分数阶慢扩散方程的一类有效差分方法

对时间分数阶慢扩散方程提出一类数值差分方法:显-隐(Explicit-Implicit, E-I)和隐-显(Implicit-Explicit, I-E)差分方法.它是将古典显式格式与古典隐式格式相结合构造出的一类有效差分格式.理论证明了格式解的存在唯一性,用傅里叶方法证明了格式的稳定性和收敛性.数值试验验证了理论分析,表明E-I格式和I-E格式在具有良好的精度且无条件稳定的情况下,计算速度比隐式格式提高了75%.从而用此格式解决分数阶慢扩散方程是可行的.     

一维非线性Schrdinger方程的两个无条件收敛的守恒紧致差分格式

本文对一维非线性Schrdinger方程给出两个紧致差分格式,运用能量方法和两个新的分析技巧证明格式关于离散质量和离散能量守恒,而且在最大模意义下无条件收敛.对非线性紧格式构造了一个新的迭代算法,证明了算法的收敛性,并在此基础上给出一个新的线性化紧格式.数值算例验证了理论分析的正确性,并通过外推进一步提高了数值解的精度.     

非线性Schrdinger方程的高精度守恒差分格式

本文首先分析线性Schrodinger方程一种高阶差分格式的构造方法,得到方程的耗散项.在此基础上对三次非线性Schrodinger方程,提出了一种精度为O(r2 h2)的差分格式,证明了该格式保持了连续方程的两个守恒量,且是收敛的与稳定的.并通过数值例子与已有隐格式进行了比较,结果表明,本文格式在计算量类似的情况下,提高了数值精度.     

迎风紧致格式与驱动方腔流动问题的直接数值模拟

本文给出了一种求解不可压缩流动问题的高精度差分格式,即迎风紧致格式.出发方程采用二维非定常原始变量Naiver-Stokes方程组.在差分方程中,对流项采用三阶精度的迎风紧致差分,其余空间导数项采用四阶紧致差分.本文利用该差分格式在等距网格上数值模拟了驱动方腔流动中的分离涡运动.在257×257的细网格上,Re数最高计算到10000.Re≤5000时的计算结果与前人结果符合得很好.当Re≥7500时发现流动不存在定常层流解而为非定常周期性解,并首次给出了非定常解的结果。     

一种半隐式有限体积—有限元方法的收敛性

本文研究非线性对流扩散问题的一种半隐式有限体积和有限元方法相结合的数值方法,给出数值解的收敛性及其证明。     

Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation

We present a sixth-order explicit compact finite difference scheme to solve the three-dimensional (3D) convection-diffusion equation. We first use a multiscale multigrid method to solve the linear systems arising from a 19-point fourth-order discretization scheme to compute the fourth-order solutions on both a coarse grid and a fine grid. Then an operator-based interpolation scheme combined with an extrapolation technique is used to approximate the sixth-order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid-independent convergence rate for solving convection-diffusion equations with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth-order compact (SOC) scheme, compared with the previously published fourth-order compact (FOC) scheme.     

High-order ADI method for stream-function vorticity equations

Many recent works have demonstrated the efficiency of high-order compact (HOC) difference schemes on the stream-function and vorticity formulation of 2-D incompressible Navier-Stokes equations. HOC discretizations induce cross spatial derivatives which are treated explicitly in most ADI schemes. Recently, Karaa and Zhang proposed a fourth-order ADImethod for solving convection-diffusion problems efficiently. In this work, we extend this method to the solution of incompressible Navier-Stokes equations. The driven flow in a square cavity is used as a model problem and numerical results are compared with other results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一种求解运动曲面上对流扩散方程的三维水平集方法

给出了一种求解运动曲面上对流扩散方程的三维水平集算法. 水平集函数被用来表示曲面.曲面上的微分方程及其解通过水平集方法被延拓到包含曲面的一个小邻域中. 一种半隐式的Crank-Nicholson 格式被用来做时间推进, 中心差分和三阶加权实质无振荡(WENO) 格式被分别用来离散方程中的扩散项和对流项. 分析证明了它在标准的Courant-Friedrichs-Lewy (CFL) 条件下的稳定性. 数值算例显示了它能取得二阶精度.     

Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems

Some least-squares mixed finite element methods for convection-diffusion problems, steady or nonstationary, are formulated, and convergence of these schemes is analyzed. The main results are that a new optimal a priori error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-squares mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection-diffusion equation are formulated. Also, some systematic theories on convergence of these schemes are established.

     

Revisit of Semi-Implicit Schemes for Phase-Field Equations

It is a very common practice to use semi-implicit schemes in various computations, which treat selected linear terms implicitly and the nonlinear terms explicitly. For phase-field equations, the principal elliptic operator is treated implicitly to reduce the associated stability constraints while the nonlinear terms are still treated explicitly to avoid the expensive process of solving nonlinear equations at each time step. However, very few recent numerical analysis is relevant to semi-implicit schemes, while "stabilized" schemes have become very popular. In this work, we will consider semi-implicit schemes for the Allen-Cahn equation with $general$ $potential$ function. It will be demonstrated that the maximum principle is valid and the energy stability also holds for the numerical solutions. This paper extends the result of Tang & Yang (J. Comput. Math., 34(5) (2016), pp. 471-481), which studies the semi-implicit scheme for the Allen-Cahn equation with $polynomial$ $potentials$.     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Unconditionally stable schemes for convection-diffusion problems

Convection-diffusion problems are basic ones in continuum mechanics. The main features of these problems are connected with the fact that their operators may have an indefinite sign. In this paper we study the stability of difference schemes with weights for convection-diffusion problems where the convective transport operator may have various forms. We construct unconditionally stable schemes for nonstationary convection-diffusion equations based on the use of new variables. Similar schemes are also used for parabolic equations where the operator represents the sum of diffusion operators.     

Semi-implicit Runge-Kutta schemes for the Navier-Stokes equations

The stationary Navier-Stokes equations are solved in 2D with semi-implicit Runge-Kutta schemes, where explicit time-integration in the streamwise direction is combined with implicit integration in the body-normal direction. For model problems stability restrictions and convergence properties are studied. Numerical experiments for the flow over a flat plate show that the number of iterations for the semi-implicit schemes is almost independent of the Reynolds number.     

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

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