A finite difference scheme for traveling wave solutions to Burgers equation

We construct a finite difference scheme for the ordinary differential equation describing the traveling wave solutions to the Burgers equation. This difference equation has the property that its solution can be calculated. Our procedure for determining this solution follows closely the analysis used to obtain the traveling wave solutions to the original ordinary differential equation. The finite difference scheme follows directly from application of the nonstandard rules proposed by Mickens. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 815–820, 1998     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

A fourth-order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations

A finite difference scheme for the two-dimensional, second-order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13-point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h⁴. Well-known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth-order convergence of the scheme. © 1995 John Wiley & Sons, Inc.     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. 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 schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.     

An optimal 25-point finite difference scheme for the Helmholtz equation with PML

In this paper, we present an optimal 25-point finite difference scheme for solving the Helmholtz equation with perfectly matched layer (PML) in two dimensional domain. Based on minimizing the numerical dispersion, we propose the refined choice strategy for choosing optimal parameters of the 25-point finite difference scheme. Numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion.     

An optimal 25-point finite difference scheme for the Helmholtz equation with PML

In this paper, we present an optimal 25-point finite difference scheme for solving the Helmholtz equation with perfectly matched layer (PML) in two dimensional domain. Based on minimizing the numerical dispersion, we propose the refined choice strategy for choosing optimal parameters of the 25-point finite difference scheme. Numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion.     

Asymptotic properties of solutions to two discrete airy equations

We construct two finite difference models for the Airy differential equation. In one model, the form of the complete asymptotic representation of the solution can be found. However, this is not the case for the second model which is based on the use of a nonstandard difference scheme. This scheme leads to a second-order, linear difference equation that is not of a form for which the theorems of Poincaré and Perron can be directly applied to obtain the asymptotic behavior of the solutions.     

Asymptotics of the 1/2-dimension,Discrete Airy Equation and its Approximating Differential Equation

A discrete finite difference model is constructed for the Airy equation using a nonstandard scheme formulated by Mickens and Ramadhani. The method of dominant balance is then applied to obtain a first-order difference equation for the solution that increases sufficiently fast as k→∞. We then calculate the corresponding approximating differential equation and obtain its exact solution as well as its “exact” discrete finite difference representation. The application of various symmetry operations allows the determination of the related rapidly decreasing solution and the oscillatory solutions for negative values of x k>=hk, where h=?x.     

Compact finite difference method for American option pricing

A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h   is large (h?0.1)$(h?0.1)$. Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.     

Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation

Abstract

In this paper, a Crank–Nicolson finite difference/finite element method is considered to obtain the numerical solution for a time fractional Sobolev equation. Firstly, the classical finite element method is presented. Stability and error estimation for the fully discrete scheme are rigorously established. However, the amount of calculation and computing time are too large due to many degrees of freedom of classical finite element scheme and nonlocality of fractional differential operator. And then the modified reduced-order finite element scheme with low dimensions and sufficiently high accuracy, which is based on proper orthogonal decomposition technique, is provided. Stability and convergence for the reduced-order scheme are also studied. At last, numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals

A new finite difference (FD) method, referred to as "Cartesian cut-stencil FD", is introduced to obtain the numerical solution of partial differential equations on any arbitrary irregular shaped domain. The 2^nd-order accurate two-dimensional Cartesian cut-stencil FD method utilizes a 5-point stencil and relies on the construction of a unique mapping of each physical stencil, rather than a cell, in any arbitrary domain to a generic uniform computational stencil. The treatment of boundary conditions and quantification of the solution accuracy using the local truncation error are discussed. Numerical solutions of the steady convection-diffusion equation on sample complex domains have been obtained and the results have been compared to exact solutions for manufactured partial differential equations (PDEs) and other numerical solutions.     

Exact finite difference scheme for linear differential equation with constant coefficients

An exact finite difference equation for the n-th order linear differential equation with real, constant coefficients is constructed. The exact finite difference scheme is expressed differently but equivalent to that given by Potts [3].     

提高反应—扩散方程有限差分格式的稳定性问题

This paper deals with the special nonlinear reaction-diffusion equation.The finite difference scheme with incremental unknowns approximating to the differential equation (2.1) is set up by means of introducing incremental unknowns methods.Through the stability analyzing for the scheme,it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.     

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.     

CEV跳-扩散模型中期权定价研究

考虑了CEV与Kou双指数跳-扩散组合模型中的期权定价问题.首先,运用Ito公式和期权定价的无套利原理,得到了模型下期权价格所满足的偏积-微分方程.然后,运用中心差分和Lagrange线性插值,分别对偏积-微分方程中的微分项和积分项进行离散化处理,再由Euler法,最终得了偏积-微分方程的有限差分格式,并且对差分方法的误差和收敛性进行了分析.最后数值实验验证了该算法是一个稳定且收敛的算法.     

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5- and 7-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme.     

A nonlinear nonstandard finite difference scheme for the linear time-dependent Schrödinger equation

We state and study the various limiting forms and their associated mathematical properties of a nonlinear finite difference scheme for the linear time-dependent Schrödinger partial differential equation (PDE). A formal solution to the full equation is given.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

Explicit block iterative method for the solution of the biharmonic equation

We develop an iterative algorithm for the solution of a finite difference approximation of the biharmonic equation over a rectangular region by using the explicit block iterative method, i.e., box over relaxation scheme. The 4- and 9-point explicit blocks were considered and performance results for the two algorithms are presented.© 1993 John Wiley & Sons, Inc.