High order parameter-robust numerical method for singularly perturbed reaction-diffusion problems

We present a high order parameter-robust finite difference method for singularly perturbed reaction-diffusion problems. The problem is discretized using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized Shishkin mesh. The convergence analysis is given and the method is proved to be almost fourth order uniformly convergent in maximum norm with respect to singular perturbation parameter ε. Numerical experiments are conducted to demonstrate the theoretical results.     

A defect–correction parameter-uniform numerical method for a singularly perturbed convection–diffusion problem in one dimension

Monotone second order parameter-robust numerical methods for singularly perturbed differential equations can be designed using the principles of defect-correction. However, the proofs of second order parameter-uniform convergence can be difficult, even in one dimension. In this paper, we examine a variant of the standard defect-correction scheme. This variant is proposed in order to simplify the analysis of a defect-correction method in the case of a one dimensional convection–diffusion problem. Numerical results are presented to validate the theoretical results.*This research was partially supported by the project MEC/FEDER MTM 2004-019051 and the grant EUROPA XXI of the Caja de Ahorros de la Inmaculada.     

An almost third order finite difference scheme for singularly perturbed reaction-diffusion systems

This paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction-diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given.     

A UNIFORM NUMERICAL METHOD FOR QUASILINEAR SINGULAR PERTURBATION PROBLEM WITH A TURNING POINT

A nonlinear difference scheme is given for solving a quasilinear siagularly perturbed two-point boundary value problem with a turning point. The method uses non-equidistant discretization meshes. The solution of the scheme is shown to be first order accurate in the discrete L^∞ norm, uniformly in the perturbation parameter.     

A numerical investigation of blow-up in reaction-diffusion problems with traveling heat sources

This paper studies the numerical solution of a reaction-diffusion differential equation with traveling heat sources. According to the fact that the locations of heat sources are known, we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem. Several examples are provided to demonstrate the efficiency of the new moving mesh method, especially in the case of two or three traveling heat sources. Moreover, numerical results illustrate that the speed of the movement of the heat source is critical for blow-up when there is only one traveling heat source. For the case of two traveling heat sources, blow-up depends not only on the speed but also on the distance between the two traveling heat sources.     

An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems

In this paper we develop a numerical method for two-dimensionaltime-dependent reaction-diffusion problems. This method, whichcan immediately be generalized to higher dimensions, is shownto be uniformly convergent with respect to the diffusion problems.This method, which can immediately be generalized to higherdimensions, is shown to be uniformly convergent with respectto the diffusion parameter.     

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in for . Here is the maximum side length of mesh elements, while the number of mesh nodes does not exceed . Numerical experiments are performed to support the theoretical results.

     

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.     

A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems

A coupled system of two singularly perturbed linear reaction–diffusiontwo-point boundary value problems is examined. The leading termof each equation is multiplied by a small positive parameter,but these parameters may have different magnitudes. The solutionsto the system have boundary layers that overlap and interact.The structure of these layers is analysed, and this leads tothe construction of a piecewise-uniform mesh that is a variantof the usual Shishkin mesh. On this mesh central differencingis proved to be almost first-order accurate, uniformly in bothsmall parameters. Supporting numerical results are presentedfor a test problem.     

An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

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     

Mesh‐independent convergence of the modified inexact Newton method for a second order non‐linear problem

In this paper, we consider an inexact Newton method applied to a second order non‐linear problem with higher order non‐linearities. We provide conditions under which the method has a mesh‐independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non‐linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd.     

On a system of reaction-diffusion equations describing a population with two age groups

A system of reaction-diffusion equations modelling a population divided into juvenile and adult age groups is studied. The system is not co-operative but its linear part is and this makes it possible to establish existence, non-existence and stability results for non-negative solutions of the system in terms of the principal eigenvalue of the corresponding linearized system.     

Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an -shaped domain

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional -shaped domain subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle . We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.

     

A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction-diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results.     

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.      

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.     

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     

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a₁,b₀,b₁,c₁,d₀, and d₁ are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd.