Pointwise uniformly convergent numerical treatment for the non-stationary Burger–Huxley equation using grid equidistribution

This article presents a numerical method for solving the singularly perturbed Burger–Huxley equation on a rectangular domain. That is, the highest-order derivative term in the equation is multiplied by a very small parameter. This small parameter is known as the perturbation parameter. When the perturbation parameter specifying the problem tends to zero, the solution of the perturbed problem exhibits layer behaviour in the outflow boundary region. Most conventional methods fail to capture this layer behaviour. For this reason, there is much current interest in the development of a robust numerical method that may handle the difficulties occurring due to the presence of the perturbation parameter and the nonlinearity of the problem. To solve both of these difficulties a numerical method is constructed. The first step in this direction is the discretization of the time variable using Euler's implicit method with a constant time step. This produces a nonlinear stationary singularly perturbed semidiscrete problem class. The problem class is then linearized using the quasilinearization process. This is followed by discretization in space, which uses the standard upwind finite difference operator. An extensive amount of analysis is carried out in order to establish the convergence and stability of the proposed method. Numerical experiments are carried out for model problems to illustrate graphically the theoretical results. The results indicate that the scheme faithfully mimics the dynamics of the differential equation.     

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

We introduce a high order parameter-robust numerical method to solve a Dirichlet problem for one-dimensional time dependent singularly perturbed reaction-diffusion equation. A small parameter ε is multiplied with the second order spatial derivative in the equation. The parabolic boundary layers appear in the solution of the problem as the perturbation parameter ε tends to zero. To obtain the approximate solution of the problem we construct a numerical method by combining the Crank–Nicolson method on an uniform mesh in time direction, together with a hybrid scheme which is a suitable combination of a fourth order compact difference scheme and the standard central difference scheme on a generalized Shishkin mesh in spatial direction. We prove that the resulting method is parameter-robust or ε-uniform in the sense that its numerical solution converges to the exact solution uniformly well with respect to the singular perturbation parameter ε. More specifically, we prove that the numerical method is uniformly convergent of second order in time and almost fourth order in spatial variable, if the discretization parameters satisfy a non-restrictive relation. Numerical experiments are presented to validate the theoretical results and also indicate that the relation between the discretization parameters is not necessary in practice.     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

Adaptive Fault Tolerant Control of Multi-time-scale Singularly Perturbed Systems

This paper studies the fault tolerant control, adaptive approach, for linear time-invariant two-time-scale and three-time-scale singularly perturbed systems in presence of actuator faults and external disturbances. First, the full order system will be controlled using ε-dependent control law. The corresponding Lyapunov equation is ill-conditioned due to the presence of slow and fast phenomena. Secondly, a time-scale decomposition of the Lyapunov equation is carried out using singular perturbation method to avoid the numerical stiffness. A composite control law based on local controllers of the slow and fast subsystems is also used to make the control law ε-independent. The designed fault tolerant control guarantees the robust stability of the global closed-loop singularly perturbed system despite loss of effectiveness of actuators. The stability is proved based on the Lyapunov stability theory in the case where the singular perturbation parameter is sufficiently small. A numerical example is provided to illustrate the proposed method.     

A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers

A numerical scheme for a class of two-point singularly perturbed boundary value problems with an interior turning point having an interior layer or twin boundary layers is proposed. The solution of this type of problem exhibits a transition region between rapid oscillations and the exponential behaviour. The problem with interior turning point represents a one-dimensional version of stationary convection–diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin. To solve these problems numerically, we consider a scheme which comprises quintic B-spline collocation method on an appropriate piecewise-uniform mesh, which is dense in the neighbourhood of the interior/boundary layer(s). The method is shown to be parameter-uniform with respect to the singular perturbation parameter ?. Some relevant numerical examples are illustrated to verify the theoretical aspects computationally. The results compared with other existing methods show that the proposed method provides more accurate solutions.     

Tailored Finite Point Method for a Singular Perturbation Problem with Variable Coefficients in Two Dimensions

In this paper, we propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small ε, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.     

Crank–Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection–diffusion equations

A numerical approach is proposed to examine the singularly perturbed time-dependent convection–diffusion equation in one space dimension on a rectangular domain. The solution of the considered problem exhibits a boundary layer on the right side of the domain. We semi-discretize the continuous problem by means of the Crank–Nicolson finite difference method in the temporal direction. The semi-discretization yields a set of ordinary differential equations and the resulting set of ordinary differential equations is discretized by using a midpoint upwind finite difference scheme on a non-uniform mesh of Shishkin type. The resulting finite difference method is shown to be almost second-order accurate in a coarse mesh and almost first-order accurate in a fine mesh in the spatial direction. The accuracy achieved in the temporal direction is almost second order. An extensive amount of analysis has been carried out in order to prove the uniform convergence of the method. Finally we have found that the resulting method is uniformly convergent with respect to the singular perturbation parameter, i.e. ?-uniform. Some numerical experiments have been carried out to validate the proposed theoretical results.     

Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems

In this paper, we propose a characteristic tailored finite point method (CTFPM) for solving the convection-diffusion-reaction equation with variable coefficients. We develop an algorithm to construct a streamline-aligned grid for the CTFPM. Our numerical tests show for small diffusion coefficient the CTFPM solution resolves the internal and boundary layers regardless the mesh size, and depicts that CTFPM method with a streamline grid has excellent performance compared with the tailored finite point method and a streamline upwind finite element method when ε is small.     

A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem

In this paper we address several issues arising from a singularly perturbed fourth order problem with small parameter ε. First, we introduce a new family of non-conforming elements. We then prove that the corresponding finite element method is robust with respect to the parameter ε and uniformly convergent to order h ^1/2. In addition, we analyze the effect of treating the Neumann boundary condition weakly by Nitsche’s method. We show that such treatment is superior when the parameter ε is smaller than the mesh size h and obtain sharper error estimates. Such error analysis is not restricted to the proposed elements and can easily be carried out to other elements as long as the Neumann boundary condition is imposed weakly. Finally, we discuss the local error estimates and the pollution effect of the boundary layers in the interior of the domain.     

On Parabolic Boundary Layers for Convection–Diffusion Equations in a Channel: Analysis and Numerical Applications

In this article we discuss singularly perturbed convection–diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, when this structure is available; see e.g. [Cheng, W. and Temam, R. (2002). Comput. Fluid. V.31, 453–466; Jung, C. (2005). Numer. Meth. Partial Differ. Eq. V.21, 623–648]. This approach is developed in this article for a convection–diffusion equation. Using an analytical approach, we first derive an approximate (simplified) form of the parabolic boundary layers (elements) for our problem; we then develop new numerical schemes using these boundary layer elements. The results are performed for the perturbation parameter ε in the range 10⁻¹–10⁻¹⁵ whereas the discretization mesh is in the range of order 1/10–1/100 in the x-direction and of order 1/10–1/30 in the y-direction. Indications on various extensions of this work are briefly described at the end of the Introduction.Dedicated to David Gottlieb on his 60th birthday.     

Analysis of a Penalty Method

In this article we study the simplest one-dimensional transport equation u _t +au _x =f and study the implementation of the boundary condition using a penalty method combined with a P1 finite element discretization. We discuss the convergence of the method when both the penalty parameter ? and the mesh size h go to zero, in sequence or simultaneously. Some numerical simulations are reported also showing the efficiency of the method. Numerical simulations are also made for the similar problem in space dimension?2.     

A nonconforming finite element method for a singularly perturbed boundary value problem

We analyze a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of orderh ^1/2 in a norm slightly stronger than the energy norm. Our proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.     

Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Planck's constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation term is approximated by nonstandard technique, the potential is approximated by its mean value on the cell and the complex dependent boundary conditions are handled by exact schemes. In the second scheme, the deficiency of classical schemes is corrected by the inner expansion in the boundary layer region. Numerical simulations supporting the performance of the schemes are presented.     

Some simple boundary value problems with corner singularities and boundary layers

Some boundary value problems for a second-order elliptic partial differential equation in a polygonal domain are considered. The highest order terms in the equation are multiplied by a small parameter, leading to a singularly perturbed problem. The singular perturbation causes boundary layers and interior layers in the solution, and the corners of the polygon cause corner singularities in the solution. The paper considers pointwise bounds for derivatives of the solution that show the influence of these layers and corner singularities. Several recent results on this problem are surveyed, and some open problems are stated and discussed.     

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.     

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.     

Green element computation of the Sturm-Liouville equations

A mathematical derivation of a new numerical procedure called the Green element method (GEM) is presented and applied to the solution of Sturm-Liouville problems. The GEM is a numerical technique which expands the scope of application of the boundary element method (BEM) by implementing the singular boundary integral theory in an element-by-element fashion; and like the finite element method (FEM) gives rise to a banded coefficient matrix which is easy to handle numerically. For this application, the location of both the field and the source nodes within the same element makes it possible for integrations to be carried out accurately, thereby enhancing the accuracy of discrete equations. The method is therefore easy to apply and, because of its domain based implementation, it maintains the flexibility of the FEM. We apply the GEM to the solution of boundary value differential equations which represent the form of Sturm-Liouville problems, and its capability is demonstrated by comparing the results with those of the finite element methods available in the literature. Satisfactory results and a second-order accuracy were found to be exhibited.     

Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with 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 grids. The nonuniform grids are obtained by equidistribution of a positive monitor function, which is a linear combination of a constant and the second-order spatial derivative of the singular component of the solution on every temporal level. Truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.     

Finite central difference/finite element approximations for parabolic integro-differential equations

We study the numerical solution of an initial-boundary value problem for parabolic integro-differential equation with a weakly singular kernel. The main purpose of this paper is to construct and analyze stable and high order scheme to efficiently solve the integro-differential equation. The equation is discretized in time by the finite central difference and in space by the finite element method. We prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order O(Δt ² + h ^l ). A numerical example demonstrates the theoretical results.