Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems

This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection–diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the ε-uniform accuracy of simple upwinding in the discrete supremum norm from O (N ⁻¹ ln N + Δt) to O (N ⁻² ln² N + Δt ²), where N is the number of mesh-intervals in the spatial direction and Δt is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.     

Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform meshes. The nonuniform meshes are obtained by equidistributing a positive monitor function, which depends on the second-order spatial derivative of the singular component of the solution. The truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. Semilinear IBVPs are also solved. Numerical experiments are carried out to support the theoretical results.     

An initial-value technique to solve third-order reaction–diffusion singularly perturbed boundary-value problems

The aim of this paper is to build an efficient initial-value technique for solving a third-order reaction–diffusion singularly perturbed boundary-value problem. Using this technique, a third-order reaction–diffusion singularly perturbed boundary-value problem is reduced to three approximate unperturbed initial-value problems and then Runge–Kutta fourth-order method is used to solve these unperturbed problems numerically.     

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.     

“Booster method” for singularly perturbed robin problems i

An improved numerical method is presented for singularly perturbed Robin problems in this paper. In this method (“Booster method”), an asymptotic approximation is incorporated into a finite difference scheme to improve the numerical solution. Error estimates are proposed for the present method. In order to show the efficiency numerical examples are presented.     

Higher-order convergence with fractional-step method for singularly perturbed 2D parabolic convection–diffusion problems on Shishkin mesh

In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., $O(N^{?1+\beta }lnN+\Delta t)$. To enhance the order of convergence to $O(N^{?2+\beta }{ln}^{2}N+\Delta t^2)$, we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique.     

Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layers

The present study is devoted to the numerical study of boundary value problems for singularly perturbed linear second-order differential–difference equations with a turning point. The points of the domain where the coefficient of the convection term in the singularly perturbed differential equation vanishes are known as the turning points. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behaviour depending upon the nature of the coefficient of convection term and the reaction term. In particular, this paper focuses on problems whose solution exhibits interior layers. In the development of numerical schemes for singularly perturbed differential–difference equations with a turning point, we use El-Mistikawy–Werle exponential finite difference scheme with some modifications. Some priori estimates have been established and parameter uniform convergence analysis of the proposed scheme is also discussed. Several examples are considered to demonstrate the performance of the proposed scheme and effect of the size of the delay/advance arguments and coefficients of the delay/advance term on the layer behaviour of the solution.     

A parameter uniform numerical method for a system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients

In this paper, a weakly coupled system of two singularly perturbed convection-diffusion equations with discontinuous convection coefficients is examined. A finite difference scheme on Shishkin mesh generating the parameter uniform convergence in the global maximum norm is constructed for solving this problem. Numerical results which are in agreement with the theoretical results are presented.     

A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms

In this article, a parameter-uniform hybrid numerical method is presented to solve a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms. Due to these discontinuities, interior layers appear in the solution of the problem considered. The hybrid numerical method uses the standard finite difference scheme in the coarse mesh region and the cubic spline difference scheme in the fine mesh region which is constructed on piecewise-uniform Shishkin mesh. Second order one sided difference approximations are used at the point of discontinuity. Error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to validate the theoretical results.     

An ϵ-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem

In this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ?-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.     

Singularly perturbed reaction–diffusion equations in a circle with numerical applications

The goal of this article is to study the boundary layers of reaction–diffusion equations in a circle and provide some numerical applications which utilize the so-called boundary layer elements. Via the boundary layer analysis, we obtain the valid asymptotic expansions at any order and devise boundary layer elements to be conveniently used in the finite element schemes. Using boundary layer elements incorporated in the finite element space, we obtain accurate numerical solutions in a quasi-uniform mesh with convergence of order 2.     

High-order time stepping Fourier spectral method for multi-dimensional space-fractional reaction–diffusion equations

This paper introduces a high-order time stepping scheme, that is based on using Fourier spectral in space and a fourth-order diagonal Padé approximation to the matrix exponential function for solving multi-dimensional space-fractional reaction–diffusion equations. The resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non-linear system at each time step while maintaining the stability of the scheme. The non-locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices. This scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator. It also attains spectral convergence for multiple spatial dimensions. The stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions, which provides an indication of the stability of the method. The convergence analysis is performed empirically to show that the scheme is fourth-order accurate in time, as expected. Numerical experiments on reaction–diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space-fractional reaction–diffusion equations and to validate the effectiveness of the scheme.     

Asymptotic initial-value method for a system of singularly perturbed second-order ordinary differential equations of convection–diffusion type

In this paper, a numerical method is suggested to solve a class of boundary value problems (BVPs) for a weakly coupled system of singularly perturbed second-order ordinary differential equations of convection–diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the BVP is constructed by using the basic ideas of a well known perturbation method namely Wentzal, Kramers and Brillouin (WKB). Then, some initial value problems (IVPs) are constructed such that their solutions are the terms of this asymptotic expansion. These problems happen to be singularly perturbed problems and, therefore, exponentially fitted finite difference schemes are used to solve these problems. As the BVP is converted into a set of IVPs and an asymptotic expansion approximation is used, the present method is termed as asymptotic initial-value method. The necessary error estimates are derived and examples provided to illustrate the method.     

“booster method” for singularly perturbed robin problems-II

Singularly perturbed Robin problems are considered in this paper. In order to get an improved numerical solution to these problems, a computational method (“Booster Method”) is suggested. In this method, an asymptotic approximation is incorporated into a finite difference scheme to improve the numerical solution. Error estimates are proposed when implemented in known difference schemes. Numerical examples are presented to illustrate the present method.     

The SDFEM for singularly perturbed convection–diffusion problems with discontinuous source term arising in the chemical reactor theory

In this paper, we consider singularly perturbed boundary-value problems for second-order ordinary differential equations with discontinuous source term arising in the chemical reactor theory. A parameter-uniform error bound for the solution is established using the streamline-diffusion finite-element method on piecewise uniform meshes. We prove that the method is almost second-order convergence for solution and first-order convergence for its derivative in the maximum norm, independently of the perturbation parameter. Numerical results are provided to substantiate the theoretical results.     

Discontinuous Legendre wavelet Galerkin method for reaction–diffusion equation

This paper proposes a novel numerical method, that is, discontinuous Legendre wavelet Galerkin technique for solving reaction–diffusion equation (RDE). Specifically, variational formulation and corresponding numerical fluxes of this type equation are devised by utilizing the advantages of both Legendre wavelet bases and discontinuous Galerkin approach. Furthermore, adaptive algorithm, stability and error analysis of this method have been discussed. Especially, the distinctive features of the presented approach are easy to cope with a variety of boundary conditions and able to effectively approximate solution of the RDE with less execution and storage space. Finally, numerical tests affirm better accuracy for a range of benchmark problems and demonstrate the validity and utility of this approach.     

Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection–diffusion equations

In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection–diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.     

An ϵ-uniform fitted operator method for solving boundary-value problems for singularly perturbed delay differential equations: Layer behavior

In this paper, an ?-uniform fitted operator method which solves boundary-value problems for singularly perturbed differential-difference equations containing small delay with boundary layer behavior is presented. Both the cases, i.e., when boundary layer is on the left side and when boundary layer is on the right side are discussed here. It is shown that the scheme is ?-uniform by establishing the error estimate. The effect of small delay on the boundary layer solution is shown by considering several numerical experiments. Numerical results in terms of maximum errors are tabulated and plots giving computed and exact solution demonstrate the efficiency of the method.