Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions

In this paper, we present a numerical approach to a class of nonlinear reaction-diffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this work we construct and analyze some finite difference schemes used to solve a class of time‐dependent one‐dimensional convection‐diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank‐Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N^?2log²N in space, if the Crank‐Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A‐stable SDIRK with two stages and a third‐order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

On the compact difference scheme for the two-dimensional coupled nonlinear Schrödinger equations

A high-order finite difference method for the two-dimensional coupled nonlinear Schrödinger equations is considered. The proposed scheme is proved to preserve the total mass and energy in a discrete sense and the solvability of the scheme is shown by using a fixed point theorem. By converting the scheme in the point-wise form into a matrix–vector form, we use the standard energy method to establish the optimal error estimate of the proposed scheme in the discrete L²-norm. The convergence order is proved to be of a fourth-order in space and a second-order in time, respectively. Finally, some numerical examples are given in order to confirm our theoretical results for the numerical method. The numerical results are compared with exact solutions and other existing method. The comparison between our numerical results and those of Sun and Wangreveals that our method improves the accuracy of space and time directions.     

Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes

In this article, we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonarbitrage liquidity model with observable parameters for derivatives. This nonlinear model considers that the parameters involved are observable from order book data. The proposed numerical method use a exact difference scheme in the linear convection‐reaction term, and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. To illustrate the accuracy of the method, the numerical results are compared with those produced by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 210‐221, 2014     

A difference scheme for a parabolic system modelling the thermoelastic contacts of two rods

In this article, we study a sequence of finite difference approximate solutions to a parabolic system, which models two dissimilar rods that each rod is fixed at one end and is free to expand or contact at the other end. A finite difference scheme is derived by the method of reduction of order on nonuniform mesh. The unique solvability, unconditional stability, and convergence of the difference scheme are proved. The convergence order is of order two in both time and space. The convergence of iterative algorithm for the difference scheme are also discussed. A numerical example is presented to demonstrate the theoretical results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A FINITE DIFFERENCE METHOD FOR THE HEAT EQUATION WITH A NONLINEAR BOUNDARY CONDITION

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

热方程非线性边界条件问题的有限差分方法

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

Numerical Approximation and Error Analysis for the Timoshenko Beam Equations with Boundary Feedback

In this paper,the numerical approximation of a Timoshenko beam with bound- ary feedback is considered.We derived a linearized three-level difference scheme on uniform meshes by the method of reduction of order for a Timoshenko beam with boundary feedback.It is proved that the scheme is uniquely solvable,unconditionally stable and second order convergent in L_∞norm by using the discrete energy method. A numerical example is presented to verify the theoretical results.     

On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation

In this paper, we study the initial-boundary value problem of the usual Rosenau-RLW equation by finite difference method. We design a conservative numerical scheme which preserves the original conservative properties for the equation. The scheme is three-level and linear-implicit. The unique solvability of numerical solutions has been shown. Priori estimate and second order convergence of the finite difference approximate solutions are discussed by discrete energy method. Numerical results demonstrate that the scheme is efficient and accurate.     

自然对流换热问题基于混合元法的差分格式及其数值模拟

研究自然对流换热问题,通过对于空间变量采用有限元离散而对于时间变量用差分离散,导出一种基于混合有限元法的最低阶的差分格式,这种格式可以同时求出流体的速度、温度和压力的数值解,并给出了模拟方腔流的自然换热的数值例子。     

Two finite difference schemes for the phase field crystal equation

The phase field crystal(PFC) model is a nonlinear evolutionary equation that is of sixth order in space.In the first part of this work,we derive a three level linearized difference scheme,which is then proved to be energy stable,uniquely solvable and second order convergent in L_2 norm by the energy method combining with the inductive method.In the second part of the work,we analyze the unique solvability and convergence of a two level nonlinear difference scheme,which was developed by Zhang et al.in 2013.Some numerical results with comparisons are provided.     

常系数扩散方程的三层九点差分格式的稳定性（英文）

In this paper,we study a numerical solution of diffusion equation.We propose a three level-nine-point implicit difference scheme and prove the difference scheme is compatible with diffusion equation,second order convergent,unconditionally stable.A numerical experiments show,the difference scheme works well inside domain,but not near the discontinuous initial-boundary points,there are still has a vibration even though it was proved unconditionally stable theoretically.We take an action to solve the disturbance,give an Algorithm,Algorithm says,we must do some primal work at the discontinuous-initial-boundary points,then starting numerical solution according the three level-nine-point implicit difference scheme we proposed in this paper.The numerical example is done once again,and there is no disturbance or vibration,our Algorithm performed well all in domain and on the boundary points with small error and good accuracy,so the Algorithm we recommended is feasible and effective.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.     

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

Numerical simulation of a Finite Moment Log Stable model for a European call option

Compared to the classical Black-Scholes model for pricing options, the Finite Moment Log Stable (FMLS) model can more accurately capture the dynamics of the stock prices including large movements or jumps over small time steps. In this paper, the FMLS model is written as a fractional partial differential equation and we will present a new numerical scheme for solving this model. We construct an implicit numerical scheme with second order accuracy for the FMLS and consider the stability and convergence of the scheme. In order to reduce the storage space and computational cost, we use a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) to solve the discrete scheme. A numerical example is presented to show the efficiency of the numerical method and to demonstrate the order of convergence of the implicit numerical scheme. Finally, as an application, we use the above numerical technique to price a European call option. Furthermore, by comparing the FMLS model with the classical B-S model, the characteristics of the FMLS model are also analyzed.     

<Emphasis Type="Italic">ε</Emphasis>-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers

This article is devoted to the study of a hybrid numerical scheme for a class of singularly perturbed parabolic convection-diffusion problems with discontinuous convection coefficients. In general, the solutions of this class of problems possess strong interior layers. To solve these problems, we discretize the time derivative by the backward-Euler method and the spatial derivatives by a hybrid finite difference scheme (a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions) on a layer resolving piecewise-uniform Shishkin mesh. It is proved that the method converges uniformly in the discrete supremum norm with almost second-order spatial accuracy. Moreover, an optimal order of convergence (up to a logarithmic factor) is obtained inside the layer regions. Extensive numerical experiments are conducted to support the theoretical results and also, to demonstrate the accuracy of this method.     

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.