A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

拟线性抛物型方程奇异摄动问题的数值解法

本文讨论拟线性抛物型方程奇异摄动问题的差分解法,在非均匀网格上建立了线性三层差分格式,并证明了在离散的L₂范数意义下格式的一致收敛性,最后给出了一些数值例子.     

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.     

时间导数项含小参数的抛物方程的数值解法

本文讨论时间导数项含小参数的抛物方程.我们依Бахволов构造非均匀网格的差分格式,并证明了格式的一阶一致收敛性.给出了数值结果.     

Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid

A singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. A first-order rate of convergence, independent of the perturbation parameter, is established by using the theory of the discrete Green's function. Unlike some previous analysis for the fully discretized approach, the present problem does not require the conservative form of the underlying boundary value problem. Dedicated to Dr. Charles A. Micchelli for his 60th birthday Mathematics subject classifications (2000) 65L10, 65L12. This work is supported by National Science Foundation of China, the key project of China State Education Ministry and Hunan Education Commission.     

An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience

A class of time‐dependent singularly perturbed convection‐diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection‐diffusion problem where the second‐order derivative with respect to the spatial direction is multiplied by a small perturbation parameter and the shifts are of is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank‐Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second‐order accurate in time and almost first‐order accurate in space with respect to the perturbation parameter . To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of . The effect of the shifts on the boundary layer has also been observed.     

高阶椭圆型偏微分方程奇异摄动问题一致收敛差分格式

本文,我们讨论了一类高阶椭圆型偏微分方程奇异摄动问题。给出了连续问题解的先验估计。另外,我们还提供了一种数值求解该类问题的指数型差分格式。最后,证明了差分问题的解在能量范数意义下关于小参数一致收敛到连续问题的解。     

Domain decomposition for a parabolic convection‐diffusion problem

A two‐dimensional convection‐diffusion problem of parabolic type is considered. A multidomain decomposition algorithm with nonoverlapping subdomains based on a upwind scheme and on a piecewise equidistant mesh is investigated. Uniform in a perturbation parameter convergence properties of the algorithm are established. Numerical experiments complement the theoretical results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results. This revised version was published online in June 2006 with corrections to the Cover Date.     

非线性常微分方程转向点问题的数值解

本文利用文[3]的技巧得到了具转向点的非线性常微分方程边值问题的导数估计,再结合文[4]的方法,证明了所构造的差分格式关于小参数ε的一致收敛性.我们给出了数值例子,数值结果与理论分析完全符合.     

在非均匀网格上解椭圆—双曲型偏微分方程奇异摄动问题

本文考察了椭圆一双曲型偏微分方程奇异摄动问题(1.1),证明了迎风差分格式在一特殊的非均匀网格上是一阶一致收敛的.最后给出了一些数值结果.     

奇异摄动初值问题的一致收敛有限差分法

通过一阶和二阶导数讨论了带小参数的线性二阶常微分方程的初值问题.在均匀网格上得出了带常数拟合因子的指数型拟合差分格式,给出了离散最大模意义上的一阶一致收敛性.文中给出了数值结果.     

Graded Galerkin methods for the high‐order convection‐diffusion problem

We develop a Galerkin method using the Hermite spline on an admissible graded mesh for solving the high‐order singular perturbation problem of the convection‐diffusion type. We identify a special function class to which the solution of the convection‐diffusion problem belongs and characterize the approximation order of the Hermite spline for such a function class. The approximation order is then used to establish the optimal order of uniform convergence for the Galerkin method. Numerical results are presented to confirm the theoretical estimate.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

Numerical analysis of a model of cure process for composites

We consider a composite material composed of fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. In this paper, we are interested in the computation of approximate solutions. We propose a family of discretized problems depending on two parameters (β₁, β₂) ε [0, 1]² which split the linear and non‐ linear terms in implicit and explicit parts. We prove the stability and convergence of the discretization for any (β₁, β₂) ε [½, 1 ] × [0, 1]. We present also some numerical results. Copyright © 2006 John Wiley & Sons, Ltd.     

On the non-existence of -uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems

In this paper fitted finite difference methods on a uniform mesh with internodal spacing , are considered for a singularly perturbed semilinear two-point boundary value problem. It is proved that a scheme of this type with a frozen fitting factor cannot converge -uniformly in the maximum norm to the solution of the differential equation as the mesh spacing goes to zero. Numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors.

     

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014     

Uniform Convergence Analysis of Finite Difference Scheme for Singularly Perturbed Delay Differential Equation on an Adaptively Generated Grid

<正>Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.     

半线性常微分方程混合边值奇摄动问题的一致差分格式*

本文应用分离奇性法研究半线性常微分方程混合边值奇摄动问题的一致差分格式,我们证明了所构造的I1'in型差分格式关于小参数ε的一阶一致收敛性.在本文的最后,我们给出一个数值例子,计算结果与理论分析相符合.     

半线性抛物型方程奇异摄动问题的数值解

本文讨论具有抛物边界层的半线性抛物型方程奇异摄动问题的数值解法,在非均匀网格上构造了两层非线性差分格式,证明了差分格式是一致收敛的,给出了一些数值例子.