The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

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. +e solution is expressed as u � w + v, where w is the solution of corresponding auxiliary boundary value problem and v is a singular correction with direct expressions. +e 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. +e 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.


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]. ese 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. e 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 reactiondiffusion type with discontinuous source term in [4,5], respectively. e 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 piecewiseuniform 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. e second-order singularly perturbed reaction-diffusion boundary value problem with discontinuous source term is considered: where 0 < ε ≪ 1 is a small positive parameter, c > 0. Assuming the function f(x) is sufficiently smooth in Ω, a jump discontinuity is expected at d ∈ Ω denoted by In general, this discontinuity of f gives rise to interior layers in the first derivative of the exact solution. Here, Ω − � (0, d), Ω + � (d, 1). e rational spectral collocation method was proposed in the literature [10]. A conformal map maps the collocation points clustered near the poles of [− 1, 1], which means more points are near the poles. e 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 secondorder singularly perturbed reaction-diffusion problem with nonsmooth source term. is paper is organized as follows. e asymptotic analysis and sinh transform is outlined in Section 2. e 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. e 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.

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).

Proof.
is theorem is proved constructively. Let u 1 and u 2 be particular solutions of the following differential equations, respectively: Consider the function where ϕ 1 (x) and ϕ 2 (x) are the solutions of the boundary value problems: and A and B are constants to be chosen so that u ∈ C 1 (Ω).
Using the fact that the maximum principle states that the maximum and minimum values taken on by ϕ i ∈ Ω, i � 1, 2 must occur on zΩ, then we have 0 < ϕ i < 1, i � 1, 2, and We need to choose the constants A and B so that u ∈ C 1 (Ω); then, we impose Additionally, the constants A and B satisfy where Since the system of linear equation (7) has a solution.
□ Lemma 1. Let us suppose that a function u(x) satisfies Proof. Let x 0 be such that u(x 0 ) � min Ω u(x). If u(x 0 ) > 0, the conclusion is clearly correct. Suppose that u(x 0 ) < 0; it is clear that either x 0 ∈ Ω − ∪ Ω + or x 0 � d. In the first case, u ′ (x 0 ) � 0, u ″ (x 0 ) ≥ 0 and which is a contradiction. In the second case, the proof depends on whether or not u is differentiable at d.
and similarly, there also exists x 3 ∈ Ω h : us, we have which is a contradiction. e direct application of the maximum principle is the following stability result. □ Theorem 2. Let u(x) be the solution of (1), which can be estimated as Proof. Setting M � max(|α|, |β|, (|f(x)|/b)), we can con- Furthermore, the solution u of (1) has first-order continuous differentiability, that is, It follows from the maximum principle that v ≥ 0 for all x ∈ Ω; thus, we obtain Proof.
e existence of problem (1) has been proved in eorem 1. Let u 1 (x), u 2 (x) be the solution of problem (1), and en, we have It follows at once from eorem 2, and we can obtain us, problem (1) has a unique solution.  (1) is considered as the coupling of the following two subproblems: where c is the value of u at d, which is determined by e Shishkin decomposition method splits the solution of the singular perturbation problem (1) into regular ω and layer v components. e regular component ω is defined as the solution of For each integer k satisfying 0 ≤ k ≤ 4, the regular term ω and the singular term v satisfy the bounds where C is a constant independent of ε and Proof. is is found using the stability result in eorem 2 and the techniques in [3]. □ Lemma 3 (see [12]). Considering the singularly perturbed reaction-diffusion , the solution u of this problem has the following asymptotic expansion: where and the following inequality is satisfied: where C is a generic constant. Lemma 3 indicates that the boundary layer region of the singularly perturbed reaction-diffusion problem with smooth source term is [0, τ 0 ] and [1 − τ 0 , 1]. In other words, the position of the boundary layer is at the two endpoints in the interval [0, 1], and its width is δ � τ 0 .

Rational Spectral Collocation Method in the Barycentric
Form. A rational interpolation function p N (x) in the barycentric form of a function u(x) can be expressed as follows [15]: where ω k N k�0 are barycentric weights and x k N k�0 are distinct interpolation points. Most important in practice are the so-called Chebyshev-Gauss-Lobatto points x k � − cos(kπ/N), k � 0, 1, . . . , N, and in this case, the barycentric weights are as follows [15]: Theorem 4 (see [16]). Let D 1 and D 2 be domains in C containing J � [− 1, 1] and a real interval I, respectively. Let g: D 1 ⟶ D 2 be a conformal map such that g(J) � I. If f: D 2 ⟶ C is such that f ∘ g: D 1 ⟶ C is analytic inside and on an ellipse with foci ±1, semimajor axis length L, and semiminor axis length l, then the rational function p N (x), which interpolates f at the transformed Chebyshev points uniformly for all x ∈ [− 1, 1]. As suggested in (31), the convergence rate of the rational spectral collocation method mainly depends on the analytic region of u in the complex plane. us, the conformal map g could be chosen to enlarge the ellipse of analyticity of u ∘ g.
en, compared with the Chebyshev spectral method, a better approximation of u 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 nth derivative of the rational interpolating function p N (x) evaluated at the point x j can be expressed in the following form: where D (n) jk is the entry of the nth order differentiation matrix.
e entries of the first-and second-order differentiation matrices are given as follows [15]: Note that differentiation matrices (33) only rely on weights ω k and the points x k , which is the reason why the underlying equation does not need to be converted to new coordinates after maps.

e Sinh Transform.
To approximate the rapid changes in the boundary layer region, Tee and Trefethen have constructed the conformal map [17]: where λ, μ are the location and width of the boundary layers, respectively. e transformed Chebyshev points g λ,δ (x k ) N k�0 are clustered near the location of boundary layer x � λ, 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 g at x � 0 are continuous to preserve the spectral accuracy. For the reaction-diffusion type, the parameter in (35) should be chosen as μ � 2τ 0 .

The Rational Spectral Collocation with a
Singularly-Separated Method e 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.
e Singularity-Separated Technique. Consider the problem (1). Taking the left subproblem (19) as an example, the efficient numerical method can be constructed. e homogenous equation L − u � 0 has two eigenvalues: Let u 0 be a special solution of L − u � f; its general solution is (37) Note that if ε � 10 − k and k ≥ 4, ϕ 1 (1) � ϕ 2 (0) � e λ 2 ≈ e (− 1/ � ϵ √ ) ≥ e − 10 2 ≈ 10 − 44 � 0 can be neglected. us, the solution u l (x) of (19) can be decomposed into two parts: u l (x) � ω l (x) + v l (x), in which ω l (x) � u 0 (x) is the regular term and v l (x) � C 1 ϕ 1 (x) + C 2 ϕ 2 (x) is the singular term. e regular term ω l (x) is the solution of an auxiliary boundary value problem: It is known from the boundary conditions of equation Similarly, the solution u r (x) of (20) can also be decomposed into two parts: u r (x) � ω r (x) + v r (x). e regular term ω r (x) is the solution of an auxiliary boundary value problem: In the same way, (19) and (20) are where w l (x) and w r (x) are the solutions of the auxiliary in (38) and (39).
Proof. According to the boundary condition of (1), it is known that We can obtain C l 1 � α − (f − (0)/c), C r 2 � β − (f + (1)/c). On the contrary, the solution of equation (1) Furthermore, us, the parameters can be obtained: us, the solution of the singularly-separated technique can be obtained as where the regular terms w l (x) and w r (x) are the solutions of the auxiliary in (38) and (39) and the singular terms v l (x) and v r (x) are defined as (40).

RSC-SSM Method.
e key to the success of the RSC-SSM is obtaining a high-precision numerical solution for weakened singular perturbation problems (38). e rational spectral collocation with a sinh transform is selected.
Let the transformed Chebyshev collocation points be where g is shown in (35).
where W l � w 0 , w 1 , . . . , w N T , with w k evaluating w t k , and E N+1 is a unit matrix of N + 1-order. Let A � − (4/d 2 )εD (2) + cE N+1 , then equation (49) can be rewritten as compare the results with other existing methods. We verify the theoretical results obtained in the previous section through numerical experiments. e maximum relative errors of the solution are given by where u N and u are the numerical and exact solutions, respectively. In our computations, all experiments are carried out using MATLAB (version R2014a) on a PC with a 2.5 GHz central processing unit (Intel Core i5-2450M), 4.00 GB memory, and Windows 7 operating system.

Mathematical Problems in Engineering
Example. Consider the following singularly perturbed problem with discontinuous source term: where  Mathematical Problems in Engineering e exact solution of this problem can be expressed as where A and B are constants related to ε: Obviously, there is an interior layer at x � 0.5 in the solution.
In the RSC-SSM method, the parameters in combined transform of (35) are chosen as μ � − 4εlnε.
e RSC-SSM method and the different schemes in [9,18] are used to solve this problem with several choices of N and ε. e maximum relative errors of the three methods are listed in Table 1.
Compared with the existing difference schemes [9,18], the RSC-SSM accuracy has been significantly improved for the case with ε 2 ≤ 10 − 3 . However, when the parameters ε 2 ≥ 10 − 2 , the accuracy of the RSC-SSM is similar to that of the difference method. is is because that the discarded item reached e − (1/ε) � e − 10 ≈ 4.5 × 10 − 5 in singularly-separated technique. In Table 1, we investigate the effects of the singular perturbation parameter ε on the errors. We can see that a decrease in ε reduces the maximum relative errors. Figure 1 depicts the numerical solution and the exact solution of the whole region with different ε. e numerical solution and exact solutions in both the interior and the boundary layers are given in Figure 2. Figure 3 shows the maximum relative errors in the semilog scale with ε � 10 − 1 , 10 − 2 , 10 − 3 , and 10 − 4 , respectively. It demonstrates the influence of points N on the errors. e convergence rates are almost balanced, and even the errors have increased due to the cumulative effects with an increase in N. Figure 4 shows the pointwise errors of the function in the whole region with ε � 10 − 1 , 10 − 2 , 10 − 3 , and 10 − 4 , respectively. e smaller the parameter ε is, the more accurate the RSC-SSM result. Figures 3 and 4 also show that the error mainly comes from the discarded items in the process of singular separation.
is confirms the results discussed in the previous section.

Conclusions
In this paper, a novel numerical method, named RSC-SSM, has been proposed to solve singularly perturbed boundary value problems with discontinuous source term. e solution is composed of the weaker singularity auxiliary solution and a singular correct function. e numerical experiment illustrates that the proposed RSC-SSM is obviously a better choice to other existing numerical methods available for the singularly perturbed boundary value problems with discontinuous source term. e numerical results demonstrate the almost spectral accuracy of the proposed algorithm and coincide with the theoretical analysis. Moreover, the theoretical and numerical frameworks presented in this paper can be extended to more complex problems.

Data Availability
e data used to support the findings of this study are included in the article.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.