Abstract

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as , where is the solution of corresponding auxiliary boundary value problem and is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.

1. Introduction

Singular perturbation problems arise in mathematical modeling of physical and engineering problems, such as the boundary layer of fluid mechanics, the turning point of quantum mechanics, and the flow of large Reynolds numbers. In recent decades, the singular perturbation problem has received extensive attention. Detailed theories and analysis of the singular perturbation problem can be found in the literature [1, 2]. These problems have steep gradients in narrow layers, which is a serious obstacle in calculating classical numerical methods.

Many studies of singularly perturbed problems with nonsmooth data have been researched. The first-order Schwarz method with uniform parameters on Shishkin’s mesh was proposed for a singularly perturbed reaction-diffusion problem with a nonsmooth source term [3].Chandru and Shanthi applied a boundary value technique and hybrid difference scheme for singularly perturbed boundary value problem of reaction-diffusion type with discontinuous source term in [4, 5], respectively. The second-order finite element method was presented on a Bakhvalov–Shishkin’s mesh for singularly perturbed problems with the interior layer [6]. A uniformly convergent difference scheme was employed for singularly perturbed semilinear problems with a discontinuous source term and proved that this scheme was the first-order convergence [7]. A uniformly numerical convergence scheme based on piecewise-uniform Shishkin’s meshes was developed for a reaction-diffusion equation with a discontinuous diffusion coefficient and proved to have second-order convergence [8]. A hybrid difference scheme on a Shishkin’s mesh was introduced for a singularly perturbed reaction-diffusion problem with one and two parameters [9] with interior layers.

The second-order singularly perturbed reaction-diffusion boundary value problem with discontinuous source term is considered: where is a small positive parameter, . Assuming the function is sufficiently smooth in , a jump discontinuity is expected at denoted by . In general, this discontinuity of gives rise to interior layers in the first derivative of the exact solution. Here, .

The rational spectral collocation method was proposed in the literature [10]. A conformal map maps the collocation points clustered near the poles of , which means more points are near the poles. The parameters of the mapping depend on the position and width of the boundary layer. A rational spectral collocation in a barycentric form with sinh transform (RSC-sinh method) is applied to solve a coupled system of singularly perturbed problems and third-order singularly perturbed problems [11, 12].

To weaken the singularity and improve the accuracy of numerical simulation, the singularity-separated method (SSM) for the singular perturbation problem with constant coefficients was proposed by Chen and Yang [13]. Finite element methods with SSM were used to solve a singular perturbation problem with a single boundary layer. A novel rational spectral collocation method is presented combined with the singularity-separated technique for a system of singularly perturbed boundary value problems [14].

A novel numerical method based on the rational spectral collocation in a barycentric form with a singularly-separated method (RSC-SSM) is proposed for solving the second-order singularly perturbed reaction-diffusion problem with nonsmooth source term.

This paper is organized as follows. The asymptotic analysis and sinh transform is outlined in Section 2. The algorithmic details of RSC-SSM for second-order singularly perturbed boundary value problem are considered, and the error estimates for the method are discussed in Section 3. The singularly perturbed reaction-diffusion problem with discontinuous source term is solved in Section 4, which supports theoretical results and provides a favorable comparison with existing methods. Finally, the conclusions are drawn in Section 5.

2. Preliminaries

In this section, some useful lemmas are given, which provide information about the boundary and interior layers occurring in the solution of singularly perturbed problem (1).

Theorem 1. The problem (1) has a solution .

Proof. This theorem is proved constructively. Let and be particular solutions of the following differential equations, respectively:Consider the function where and are the solutions of the boundary value problems:and and are constants to be chosen so that . Using the fact that the maximum principle states that the maximum and minimum values taken on by must occur on , then we have , and We need to choose the constants and so that ; then, we imposeAdditionally, the constants A and B satisfywhereSince the system of linear equation (7) has a solution.

Lemma 1. Let us suppose that a function satisfies and , then .

Proof. Let be such that . If , the conclusion is clearly correct. Suppose that ; it is clear that either or . In the first case, andwhich is a contradiction. In the second case, the proof depends on whether or not is differentiable at . If is not differentiable at , then . Since , it is clear that , which is a contradiction. If is differentiable at , according to the mean value theorem, . Since , it follows that there exists a neighbourhood such that . Choose a point such that . Then, the mean value theorem indicates that :and similarly, there also exists :Thus, we havewhich is a contradiction.
The direct application of the maximum principle is the following stability result.

Theorem 2. Let be the solution of (1), which can be estimated as

Proof. Setting , we can construct satisfyingFurthermore, the solution of (1) has first-order continuous differentiability, that is,It follows from the maximum principle that for all ; thus, we obtain

Theorem 3. The problem (1) has a unique solution .

Proof. The existence of problem (1) has been proved in Theorem 1. Let be the solution of problem (1), and ; set . Then, we haveIt follows at once from Theorem 2, and we can obtain .
Thus, problem (1) has a unique solution.

2.1. Asymptotic Expansion Approximation

For the construction of the RSC-SSM method, it is necessary to use the exact solution of the problem to analyze the position and width of the boundary layer. The problem (1) is considered as the coupling of the following two subproblems: where is the value of at , which is determined by .

The Shishkin decomposition method splits the solution of the singular perturbation problem (1) into regular and layer components. The regular component is defined as the solution of

The singular component is given by

Lemma 2. For each integer satisfying , the regular term and the singular term satisfy the boundswhere is a constant independent of and

Proof. This is found using the stability result in Theorem 2 and the techniques in [3].

Lemma 3. (see [12]). Considering the singularly perturbed reaction-diffusionwhere is sufficiently smooth in , the solution of this problem has the following asymptotic expansion:whereand the following inequality is satisfied:where is a generic constant.
Lemma 3 indicates that the boundary layer region of the singularly perturbed reaction-diffusion problem with smooth source term is and . In other words, the position of the boundary layer is at the two endpoints in the interval , and its width is .

2.2. Rational Spectral Collocation Method in the Barycentric Form

A rational interpolation function in the barycentric form of a function can be expressed as follows [15]:where are barycentric weights and are distinct interpolation points. Most important in practice are the so-called Chebyshev–Gauss–Lobatto points , and in this case, the barycentric weights are as follows [15]:

Theorem 4. (see [16]). Let and be domains in containing and a real interval , respectively. Let be a conformal map such that . If is such that is analytic inside and on an ellipse with foci , semimajor axis length , and semiminor axis length , then the rational function , which interpolates at the transformed Chebyshev points satisfiesuniformly for all .
As suggested in (31), the convergence rate of the rational spectral collocation method mainly depends on the analytic region of in the complex plane. Thus, the conformal map could be chosen to enlarge the ellipse of analyticity of . Then, compared with the Chebyshev spectral method, a better approximation of could be obtained.
An advantage of the rational function in barycentric form is that its derivatives can be calculated directly using differentiation formulae instead of repeatedly using the differential quotient rule. The th derivative of the rational interpolating function evaluated at the point can be expressed in the following form:where is the entry of the th order differentiation matrix. The entries of the first- and second-order differentiation matrices are given as follows [15]:Note that differentiation matrices (33) only rely on weights and the points , which is the reason why the underlying equation does not need to be converted to new coordinates after maps.

2.3. The Sinh Transform

To approximate the rapid changes in the boundary layer region, Tee and Trefethen have constructed the conformal map [17]:where are the location and width of the boundary layers, respectively. The transformed Chebyshev points are clustered near the location of boundary layer , and the density is determined by the width of the boundary layer.

To better distinguish the singular perturbation problem with two boundary layers, Tee proposed the combined sinh transform as

All derivatives of the piecewise map at are continuous to preserve the spectral accuracy. For the reaction-diffusion type, the parameter in (35) should be chosen as .

3. The Rational Spectral Collocation with a Singularly-Separated Method

The implementation and error estimation of RSC-SSM for a singularly perturbed reaction-diffusion problem with a discontinuous source term is elaborated in this section.

3.1. The Singularity-Separated Technique

Consider the problem (1). Taking the left subproblem (19) as an example, the efficient numerical method can be constructed. The homogenous equation has two eigenvalues:

Let be a special solution of ; its general solution is

Note that if and can be neglected.

Thus, the solution of (19) can be decomposed into two parts: , in which is the regular term and is the singular term. The regular term is the solution of an auxiliary boundary value problem:

It is known from the boundary conditions of equation (38) that . The singularities of the solution are weakened.

Similarly, the solution of (20) can also be decomposed into two parts: . The regular term is the solution of an auxiliary boundary value problem:

In the same way, .

Theorem 5. The singular components of (19) and (20) arewherewhere and are the solutions of the auxiliary in (38) and (39).