色散方程的一类本性并行的差分格式

对一维色散方程给出了本性并行的一般的交替差分格式，证明了该类格式的绝对稳定性已有的交替分组显格式(AGE)是该类格式的特例．作为特例，进一步得到交替分段显一隐格式(ASF-I)和交替分段Crank-Nicolson格式(ASC-N)．数值实验比较了这几个格式数值解的精确性．     

Numerical investigation of the generalized lubrication equation

Explicit, implicit-explicit and Crank-Nicolson implicit-explicit numerical schemes for solving the generalized lubrication equation are derived. We prove that the implicit-explicit and Crank-Nicolson implicit-explicit numerical schemes are unconditionally stable. Numerical solutions obtained from both schemes are compared. Initial curves with both zero and finite contact angles are considered.     

色散方程的一类新的并行交替分段隐格式

本文给出了一组逼近色散方程的非对称差分格式，并用这组格式和对称的Crank-Nicolson型格式构造了求解色散方程的并行交替分段差分隐格式．这个格式是无条件稳定的，能直接在并行计算机上使用．数值试验表明，这个格式有很好的精度．     

KdV方程的一类本性并行差分格式

对三阶KdV方程给出了—组非对称的差分公式,并用这些差分公式和对称的Crank-Nicolson型公式构造了一类具有本性并行的交替差分格式．证明了格式的线性绝对稳定性．对—个孤立波解、二个孤立波解和三个孤立波解的情况分别进行了数值试验,并对—个孤立波解的数值解的收敛阶和精确性进行了试验和比较．     

A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.     

ALTERNATING BAND CRANK-NICOLSON METHOD FOR δu／δt=δ^2u／δx^2＋δ^2u／δy^2

the Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [1],and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2].The methods have the advantages of parallel computing,stability and good accuracy.In this paper for the two-dimensional diffusion equation,the net region is divided into bands,a special kind of block.This method is called the alternating Band Crank-Nicolson method.     

ALTERNATING BAND CRANK-NICOLSON METHOD FOR

The Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [ 1 ], and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2]. The methods have the advantages of parallel computing, stability and good accuracy. Tn this paper for the two-dimensional diffusion equation, the net region is divided into bands, a special kind of block. This method is called the alternating Band Crank-Nicolson method.     

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.     

A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

三阶非线性KdV方程的交替分段显-隐差分格式

对三阶非线性KdV方程给出了一组非对称的差分公式,用这些差分公式与显、隐差分公式组合,构造了一类具有本性并行的交替分段显-隐格式A·D2证明了格式的线性绝对稳定性．对1个孤立波解、2个孤立波解的情况分别进行了数值试验．数值结果显示,交替分段显-隐格式稳定,有较高的精确度．     

Coupled fluid-solid thermal interaction modeling for efficient transient simulation of biphasic water-steam energy systems

This paper proposes a fluid-solid coupled finite element formulation for the transient simulation of water-steam energy systems with phase change due to boiling and condensation. As it is commonly assumed in the study of thermal systems, the transient effects considered are exclusively originated by heat transfer processes. A homogeneous mixture model is adopted for the analysis of biphasic flow, resulting in a nonlinear transient advection-diffusion-reaction energy equation and an integral form for mass conservation in the fluid, coupled to the linear transient heat conduction equation for the solid. The conservation equations are approximated applying a stabilized Petrov-Galerkin FEM formulation, providing a set of coupled nonlinear equations for mass and energy conservation. This numerical model, combined with experimental heat transfer coefficients, provides a comprehensive simulation tool for the coupled analysis of boiling and condensation processes. For the treatment of enthalpy discontinuities traveling with the flow, a novel explicit-implicit time integration method based on Crank-Nicolson scheme is proposed, analyzing its accuracy and stability properties. To reduce problem size and enhance numerical efficiency, a modal superposition method with balanced truncation is applied to the solid equations. Finally, different example problems are solved to demonstrate the capabilities, flexibility and accuracy of the proposed formulation.     

Analysis of stability and error estimates for three methods approximating a nonlinear reaction-diffusion equation

We present the error analysis of three time-stepping schemes used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition. We prove $L^\infty$ stability by maximum principle arguments, and derive error estimates using energy methods for the implicit Euler, and two implicit-explicit approaches, a linearized scheme and a fractional step method. A numerical experiment validates the theoretical results, comparing the accuracy of the methods.     

High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem

We use the generalized L1 approximation for the Caputo fractional derivative, the second-order fractional quadrature rule approximation for the integral term, and a classical Crank-Nicolson alternating direction implicit (ADI) scheme for the time discretization of a new two-dimensional (2D) fractional integro-differential equation, in combination with a space discretization by an arbitrary-order orthogonal spline collocation (OSC) method. The stability of a Crank-Nicolson ADI OSC scheme is rigourously established, and error estimate is also derived. Finally, some numerical tests are given.     

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.     

Unconditional stability of parallel alternating difference schemes for semilinear parabolic systems

The general alternating schemes with intrinsic parallelism for semilinear parabolic systems are studied. First we prove the a priori estimates in the discrete H¹ space of the difference solution for these schemes. Then the existence of the difference solution for these schemes follows from the fixed point principle. Finally the unconditional stability of the general alternating schemes is proved. The alternating group explicit scheme, the alternating segment explicit–implicit scheme and the alternating segment Crank–Nicolson scheme are the special cases of the general alternating schemes.     

Burgers方程的一类交替分组方法

对于Burgers方程给出了一组新的Saul'yev型非对称差分格式，并用这些差分格式构造了求解非线性Burgers方程的交替分组四点方法．该算法把剖分节点分成若干组，在每组上构造能够独立求解的差分方程．因此算法具有并行本性，能直接在并行计算机上使用．章还证明了所给算法线性绝对稳定．数值试验表明，该方法使用简便，稳定性好，有很好的精度。     

广义非线性Schringer方程的一个新的守恒差分格式

对广义非线性Schro。d inger方程提出了一种新的差分格式.揭示了该差分格式满足两个守恒律,并证明该格式的收敛性和稳定性.数值实验结果表明,新的差分格式优于C rank-N ico lson格式以及Zhang Fei等人提出的格式.     

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L ² and H ¹ fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.     

非定常 N-S 方程的非线性 Galerkin 算法及其误差估计

本文给出了二维非定常N-S方程的三种数值格式,其中空间变量用谱非线性Galerkin算法进行离散,时间变量用有限差分离散,并研究了这些格式数值解的逼近精度．最后,给出了部分数值计算结果．