Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(α)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all α ∈ [0, 1]. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD Crank-Nicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(α) schemes provide both stable and accurate approximations for the true solution.     

Linearly implicit generalized trapezoidal formulas for nonlinear differential equations

We describe a linearlized version of the class of generalized trapezoidal formulas (GTFs) introduced in Chawla et al.[3]. For nonlinear differential equations, the obtained one-parameter class of linearly implicit generalized trapezoidal formulas (LIGTF(α)) obviate the need to solve a nonlinear system at each time step of integration, while they retain the order of accuracy and stability properties of the (functionally implicit) GTFs. The performance of the present LIGTF(α) is compared with the linearized linearly implicit trapezoidal formula (Lintrap) for nonlinear stiff ordinary differential equations (ODEs), and for nonlinear partial differential equations (PDEs) which represent nonlinear transportation-diffusion, nonlinear diffusion and nonlinear reaction-diffusion. Lintrap is known to produce unwanted oscilliations if the ratio of the time step to the spatial step becomes large. In our numerical experiments, the significance of the role played by the parameter in LIGTF(α) becomes evident in providing both stability and accuracy of the computed solution in the presence of diffusivity.     

A class of generalized trapezoidal formulas for the numerical integration of

The classical arithmetic mean (AM) trapezoidal formula is known to be A-stable. Recently, Chawla and Al-Zanaidi [1] described a modified arithmetic mean (MAM) trapezoidal formula which is L-stable. In the present paper we introduce a one-parameter family of generalized trapezoidal formulas (GTFs), which include both the AM and MAM trapezoidal formulas as special cases. A GTF can switch from an A-stable to an L-stable method depending on parameter values, and it is shown to perform, for certain selections of the parameter values, better than AM and MAM formulas for the integration of examples of the stiff systems considered.     

Block methods for parabolic equations

Block methods for the finite difference solution of linear one dimensional parabolic partial differential equations are considered. These schemes use two linear multistep formulae which, when applied simultaneously, advance the numerical solution by two time steps. No special starting procedure is required for their implementation. By careful choice of the coefficients in these formulae, all of the block methods derived in this paper are unconditionally stable and have high order accuracy. In addition, some of these schemes are suitable for problems involving a discontinuity between the initial and boundary conditions. The results of numerical experiments on two test problems are presented.     

Stability and convergence of difference schemes for multi-dimensional parabolic equations with variable coefficients and mixed derivatives

ABSTRACT

We present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results.     

Accelerating technique for implicit finite difference schemes for two dimensional parabolic partial differential equations

In this paper we define a new accurate fast implicit method for the finite difference solution of the two dimensional parabolic partial differential equations with first level condition, which may be obtained by any other method. The stability region is discussed. The suggested method is considered as an accelerating technique for the implicit finite difference scheme, which is used to find the first level condition. The obtained results are compared with some famous finite difference schemes and it is in satisfactory agreement with the exact solution.     

On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

Hartley series approximations for the parabolic equations

An approximate method for solving parabolic equations with a periodic boundary condition is proposed. The method is based upon using the Legendre series and the Hartley series to approximate the required solution. The parabolic equations are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. A numerical example is included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Tridiagonal solver-based L-stable one-step schemes for nonlinear parabolic equations

An attractive feature of the widely used Crank-Nicolson (C-N) scheme for parabolic equations is that it is a tridiagonal solver-based (TSB) scheme. But, in case of inconsistencies in the initial and boundary conditions or when the ratio of temporal to spatial steplengths is large, it can produce unwanted oscillations or an unacceptable solution. As alternative to C-N, Chawla et al. [2, 3] introduced L-stable generalized trapezoidal formulas (GTF(α)) which can give a more acceptable solution by a judicious choice of the parameter α; however, GTF are not TSB schemes. It is natural to ask for L-stable TSB schemes. In the present paper, we first introduce a one-parameter family of generalized midpoint formulas (GMF(α)); again GMF are not TSB schemes. We then introduce a two-parameter family through a linear combination of the GMF and the classical trapezoidal formula, and show the existence of a one-parameter subfamily of L-stable TSB schemes; these schemes are unconditionally stable. The computational performance of the obtained schemes is compared with the C-N scheme by considering a nonlinear reaction-diffusion equation.     

Unconditionally stable C1-cubic spline collocation method for solving parabolic equations

This article aims to present a new approach based on C¹-cubic splines introduced by Sallam and Naim Anwar [Sallam, S. and Naim Anwar, M. (2000). Stabilized cubic C¹-spline collocation method for solving first-order ordinary initial value problems, Int. J. Comput. Math., 74, 87–96.], which is A-stable, for the time integration of parabolic equations (diffusion or heat equation). The introduced method is an example of the so-called method of lines (the solution is thought to consist of space discretization and time integration), which is an extension of the 1/3-Simpson's finite-difference scheme. Our main objective is to prove the unconditional stability of the proposed method as well as to show that the method is convergent and is of order O (h ²)?+?O (k ⁴) i.e. it is a fourth-order in time and second-order in space. Computational results also show that the method is relevant for long time interval problems.     

一类非线性延迟抛物偏微分方程的Crank-Nicolson型差分格式

对一类带有Dirichlet边界条件的延迟非线性抛物偏微分方程的初边值问题建立了一个Crank-Nicolson型的线性化差分格式,用离散能量法证明了该差分格式在L_∞范数下是无条件收敛的且是稳定的,其收敛阶为O(r~2+h~2).最后,用数值算例验证了理论结果.     

随机外激非线性系统FPK方程的四阶中心C-N型隐式差分解

研究了非线性随机动力系统所对应的Fokker-Planck-kolmogorov(FPK)方程.讨论了微分方程的可朗克(Crank)一尼考尔逊(Nicolson)型隐式有限差分格式以及微分的四阶中心差分格式,将两者相结合,得到FPK方程的四阶中心C-N隐式格式差分解,并与FPK方程的精确解进行了比较.数值结果表明,该方...     

Parallel iterative methods for parabolic equations

For the problems of the parabolic equations in one- and two-dimensional space, the parallel iterative methods are presented to solve the fully implicit difference schemes. The methods presented are based on the idea of domain decomposition in which we divide the linear system of equations into some non-overlapping sub-systems, which are easy to solve in different processors at the same time. The iterative value is proved to be convergent to the difference solution resulted from the implicit difference schemes. Numerical experiments for both one- and two-dimensional problems show that the methods are convergent and may reach the linear speed-up.     

Numerical methods for two-dimensional parabolic inverse problem with energy overspecification

An inverse problem concerning the two-dimensional diffusion equation with source control parameter is considered. Four finite-difference schemes are presented for identifying the con- trol parameter which produces, at any given time, a desired energy distribution in a portion of the spatial domain. The fully explicit schemes developed for this purpose, are based on the (1,5) forward time centred space (FTCS) explicit formula, and the (1,9) FTCS scheme, are economical to use, are second-order and have bounded range of stability. Therange of stability for the 9-point finite difference scheme is less restrictive than the (1,5) FTCS formula. The fully implicit finite difference schemes employed, are based on the (5,1) backward time centred space (BTCS) formula, and the (5,5) Crank–Nicolson implicit scheme, which are unconditionally stable, but use more CPU times than the fully explicit techniques. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments are presented, and central processor (CPU) times needed for solving this inverse problem are reported.     

An alternating direction generalized trapezoidal formula scheme for parabolic differential equations in two space dimensions

A well-known ADI scheme for parabolic differential equations in two space dimensions is the Peaceman-Rachford scheme; this scheme employs the backward Euler formula for integration in time and is unconditionally stable. An ADI Crank -Nicolson scheme, which employs the classical trapezoidal formula for integration in time, is unconditionally unstable. We investigaan ADI implementation of the generalized trapezoidal formula GTF(α) for integration in time. The obtained ADI-GTF(α) scheme is unconditionally stable for all α ≥ 1; interestingly, ADIGTF(α) scheme includes the Peaceman-Rachford scheme for α→∞. Numerical experiments demonstrate that while the Peaceman-Rachford scheme can give quite pronounced unwanted oscillations in the computed solution, an ADI-GTF(α) scheme can provide a more stable and accurate approximation for the true solution.     

Multiplicative stability criteria based on difference solutions of ordinary differential equations

For computer analysis of Lyapunov stability, multiplicative criteria are proposed that are based on difference approximations to solutions of the Cauchy problem. These criteria can be applied to ordinary differential equations in normal form and include the necessary and sufficient stability conditions. For a system of linear equations with constant coefficients, information on the characteristic polynomial of the coefficient matrix and its roots is not used. The stability analysis is combined with difference solution and simulation of error accumulation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 127–142, January–February 2006.     

Stability of weak numerical schemes for stochastic differential equations

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.     

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.     

非线性RLW方程的有限差分逼近

引言 正则长波(RLW)方程是非线性长波的另一种表述形式.在进行非线性扩散波研究时,正则长波方程(RLW)因其描述大量重要的物理现象如浅水波和离子波等而占有重要的地位.     

A conservative compact difference scheme for the Zakharov equations in one space dimension

In this paper, we present a conservative fourth-order compact difference scheme for the initial-boundary value problem of the Zakharov equations. Discrete conservation laws, convergence and stability of the new scheme are proved by energy method. Several numerical results are reported to support our theoretical analysis.