Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction–Diffusion System

Abstract

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with $m<n$) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter $\epsilon \stackrel{}{\to }0$, proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results.

1. Introduction

Singularly Perturbed Problems (SPPs) are widespread in nature. SPPs often occur in different applied fields such as control systems or fluid dynamics, among others [1,2,3]. In particular, systems of Singularly Perturbed Partial Differential Equations (SPPDEs) frequently arise in the modeling of heat and mass transfer processes whenever the diffusion coefficients and the thermal conductivity are very small [4]. In [2,3], some mathematical models are developed for generators and their controls in control systems using singular perturbation techniques. The models are based on a system of n SPPs, whose first m equations (with $m<n$) have the higher order derivatives affected by a small parameter $\epsilon$. It is well-known that classical numerical methods are not satisfactory for SPPs because of the multiscale behavior of the solutions [1]. A similar type of non-SPPs has a wide range of real-time applications, for instance, the calcium distribution in nerve cells is a dynamical system that is a generalized two-dimensional space-time reaction-diffusion model [5,6].

Many unconventional numerical methods can be found in the literature to solve various types of SPPs [7,8,9,10,11]. A numerical method composed of the backward Euler method together with the HODIE (High Order via Differential Identity Expansion) scheme, classical central difference scheme, and a Shishkin mesh has appeared recently in [12] for a system of SPPDEs of Convection–diffusion (CD) type in which all the higher order spatial derivatives are multiplied by a parameter $\epsilon$. In [13], a numerical method which is a combination of the backward Euler scheme, the upwind finite difference scheme, and a nonuniform mesh based on a monitor function is devised for the aforementioned system with two equations in which the higher order spatial derivatives are multiplied by different perturbation parameters. For this same system, a numerical method consisting of the backward Euler method, a hybrid scheme, and a Shishkin mesh is developed in [14] with higher order convergence in space.

A scheme based on standard finite difference operators and the condensing mesh technique is designed in [15] to solve a system of two SPPDEs of Reaction–diffusion (RD) type in which all the higher order spatial derivatives are multiplied by a parameter ${\epsilon }^2$. The aforementioned system in which the higher order spatial derivatives are multiplied by different parameters is solved by a numerical method in [16], which consists of two additive schemes using a Shishkin mesh and a standard central difference operator. This problem is extended to a general system in [17] and solved by a numerical method composed of classical finite difference operators and appropriate layer-adapted meshes. For this same type of system with two equations, the convergence in both time and space is improved in [18] through a numerical method which consists of the Crank–Nicolson method, the classical central difference scheme, and a Shishkin mesh. The aforementioned problem is extended to a general system in [19] in which only the higher order convergence in space is maintained through a numerical method composed of classical finite difference operators and a Shishkin mesh. For the same system, the higher order convergence in time is achieved with the Crank–Nicolson method in [20].

In [21], a first-order scheme for a system of two singularly perturbed RD equations was presented, where only the second derivative of the first equation was multiplied by a small perturbation parameter. For this system, a higher order convergent scheme was developed in [22]. The article [23] deals with the construction of a numerical method for a general system of n$(n\ge 2)$ singularly perturbed RD equations in which only the first $m(m<n)$ equations are multiplied by different perturbation parameters. Inspired by [21] and [23], a numerical method is devised in [24] for a similar system of SPPDEs with more assumptions. Motivated by [2,3,15,24], in this work, a system of n SPPDEs of RD type is considered in which only the first m equations (with $m<n$) are multiplied by $\epsilon$.

The structure of the paper is as follows. In Section 2, we state the problem to be solved. The existence, uniqueness, and stability properties of the continuous solution are presented in Section 3. In addition, the decomposition of the solution (into regular and singular components) and bounds on its derivatives are established. Section 4 contains the design of a finite difference scheme using a piecewise uniform Shishkin mesh to solve the discrete problem, which provides a robust numerical approximation to the solution. In Section 5, the error analysis of the numerical solution based on barrier function techniques is derived, which sets up the theoretical results. A numerical example is given in Section 6 to conform the applicability and expected rate of accuracy of the present method with the parameter-uniform maximum pointwise error, parameter-uniform error constant, and parameter-uniform rate of convergence in the tables and graphs, which validate the major findings of the paper.

2. Continuous Problem

Let consider the singularly perturbed time-dependent RD system

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \overrightarrow{y}(x,t)}{\partial t}}-E{\displaystyle \frac{{\partial }^2\overrightarrow{y}(x,t)}{\partial x^2}}+B(x,t)\overrightarrow{y}(x,t)=\overrightarrow{f}(x,t),(x,t)\in \mathsf{\Omega },\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \mathrm{with}\overrightarrow{y}(x,0)={\overrightarrow{\psi }}_b\left(x\right)\mathrm{on}{\mathsf{\Gamma }}_b,\overrightarrow{y}(0,t)={\overrightarrow{\psi }}_l\left(t\right)\mathrm{on}{\mathsf{\Gamma }}_l,\overrightarrow{y}(1,t)={\overrightarrow{\psi }}_r\left(t\right)\mathrm{on}{\mathsf{\Gamma }}_r,\hfill \end{array}$$

(2)

where $\mathsf{\Omega }=(0,1)\times (0,T],$ $\overline{\mathsf{\Omega }}=\mathsf{\Omega }\cup \mathsf{\Gamma },$ $\mathsf{\Gamma }={\mathsf{\Gamma }}_b\cup {\mathsf{\Gamma }}_l\cup {\mathsf{\Gamma }}_r$, with ${\mathsf{\Gamma }}_b=\{(x,0):0\le x\le 1\},$ ${\mathsf{\Gamma }}_l=\{(0,t):0\le t\le T\}$ and ${\mathsf{\Gamma }}_r=\{(1,t):0\le t\le T\}.$ For $(x,t)\in \overline{\mathsf{\Omega }}$, we use the notations $\overrightarrow{y}(x,t)=$ ${(y_1(x,t),\dots ,y_n(x,t))}^T,$ $\overrightarrow{f}(x,t)=$ ${(f_1(x,t),\dots ,f_n(x,t))}^T$. We note that $B(x,t)$ and E are square matrices of dimension n with $E=diag(\epsilon ,\dots ,\epsilon ,1,\dots ,1)$, where $\epsilon$ appears m times ($m<n$), being $0<\epsilon <<1.$ It is assumed that for all $(x,t)\in \overline{\mathsf{\Omega }},$ the elements $b_{ij}(x,t),$ $i,j=1,\dots ,n,$ of $B(x,t)$ satisfy

$$\begin{array}{c}\hfill {\displaystyle b_{ij}(x,t)\le 0,\mathrm{for}i\ne j,b_{ii}(x,t)>\sum _{{}_{j\ne i}^{j=1}}^n\left|b_{ij}(x,t)\right|,}\end{array}$$

(3)

and

$$\begin{array}{c}\hfill {\displaystyle 0<\alpha <\underset{{}_{1\le i\le n}^{(x,t)\in \overline{\mathsf{\Omega }}}}{\min }\left({\displaystyle \sum _{j=1}^n{b}_{ij}(x,t)}\right),}\end{array}$$

(4)

for some constant $\alpha$. In operator form, the system (1) and (2) is written as follows:

$${\overrightarrow{L}}_{\epsilon }\overrightarrow{y}(x,t)=\overrightarrow{f}(x,t),(x,t)\in \mathsf{\Omega },$$

with ${\overrightarrow{L}}_{\epsilon }$ given by

$${\overrightarrow{L}}_{\epsilon }=I{\displaystyle \frac{\partial }{\partial t}}-E{\displaystyle \frac{{\partial }^2}{\partial x^2}}+B,$$

where I is used to denote the identity matrix.

Taking $\epsilon =0$ in (1) and (2), the reduced problem is obtained, which is given for $(x,t)\in \mathsf{\Omega }$ by

$$\begin{array}{c}{\displaystyle {\displaystyle \frac{\partial y_{0i}(x,t)}{\partial t}}+\sum _{k=1}^n{b}_{ik}(x,t)y_{0k}(x,t)=f_i(x,t),y_{0i}={\left({\overrightarrow{\psi }}_b\right)}_i\mathrm{on}{\mathsf{\Gamma }}_b,}\hfill \\ \\ {\displaystyle {\displaystyle \frac{\partial y_{0j}(x,t)}{\partial t}}-{\displaystyle \frac{{\partial }^2{y}_{0j}(x,t)}{\partial x^2}}+\sum _{k=1}^n{b}_{jk}(x,t)y_{0k}(x,t)=f_j(x,t),}\hfill \end{array}$$

(5)

for $i=1,\dots ,m$, $j=m+1,\dots ,n$, with $y_{0j}=y_j$ on $\mathsf{\Gamma }$.

In general, $y_{0i}\ne y_i$ on ${\mathsf{\Gamma }}_l\cup {\mathsf{\Gamma }}_r$ for $i=1,\dots ,m$. Hence, the components $y_i$ for $i=1,\dots ,m$ of the solution $\overrightarrow{y}$ of (1) and (2) exhibit boundary layers near ${\mathsf{\Gamma }}_l$ and ${\mathsf{\Gamma }}_r$, whereas the components $y_j$ for $j=m+1,\dots ,n$ exhibit no layer throughout the domain.

In this paper, we adopt the following notations. For any continuous vector function $\overrightarrow{\varphi }$ on $\overline{\mathsf{\Omega }},$ $\left|\right|\overrightarrow{\varphi }(x,t)\left|\right|={\max }_i\left|{\varphi }_i(x,t)\right|$ and $\left|\right|\overrightarrow{\varphi }\left|\right|={\sup }_{(x,t)\in \overline{\mathsf{\Omega }}}\left|\right|\overrightarrow{\varphi }(x,t)\left|\right|$, and for any mesh function $\overrightarrow{\mathsf{\Phi }},$ $\left|\right|\overrightarrow{\mathsf{\Phi }}(x_j,t_k)\left|\right|={\max }_i\left|{\mathsf{\Phi }}_i(x_j,t_k)\right|$ and $\left|\right|\overrightarrow{\mathsf{\Phi }}\left|\right|={\max }_{j,k}\left|\right|\overrightarrow{\mathsf{\Phi }}(x_j,t_k)\left|\right|.$ Throughout this article, $C,C_1$ and $C_2$ denote positive constants, which are independent of $x,t,\epsilon$ and the discretization parameters N and $M.$

3. Analytical Results

Some of the analytical properties, maximum principles, stability results, and derivative bounds of the exact solution (1) and (2) and its regular and singular components are estimated. The derivative bounds are necessary for the numerical analysis that is derived thoroughly in this section. The existence of the solution $\overrightarrow{y}$ of (1) and (2) is stated in Theorem 1 without proof. For more details, the reader is referred to [25].

Theorem 1.

Assume the components of B and $\overrightarrow{f}$ are sufficiently smooth functions, ${\overrightarrow{\psi }}_b\in {\left(C^4\left({\mathsf{\Gamma }}_b\right)\right)}^n,$ ${\overrightarrow{\psi }}_l\in {\left(C^2\left({\mathsf{\Gamma }}_l\right)\right)}^n,$ ${\overrightarrow{\psi }}_r\in {\left(C^2\left({\mathsf{\Gamma }}_r\right)\right)}^n$, and the following compatibility conditions are satisfied at the corners $(0,0)$ and $(1,0)$ of Γ:

$${\overrightarrow{\psi }}_b\left(0\right)={\overrightarrow{\psi }}_l\left(0\right),{\overrightarrow{\psi }}_b\left(1\right)={\overrightarrow{\psi }}_r\left(0\right),$$

(6)

$${\displaystyle \frac{d{\overrightarrow{\psi }}_l\left(0\right)}{dt}}-E{\displaystyle \frac{d^2{\overrightarrow{\psi }}_b\left(0\right)}{d{x}^2}}+B(0,0){\overrightarrow{\psi }}_b\left(0\right)=\overrightarrow{f}(0,0),$$

(7)

$${\displaystyle \frac{d{\overrightarrow{\psi }}_r\left(0\right)}{dt}}-E{\displaystyle \frac{d^2{\overrightarrow{\psi }}_b\left(1\right)}{d{x}^2}}+B(1,0){\overrightarrow{\psi }}_b\left(1\right)=\overrightarrow{f}(1,0),$$

(8)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d^2{\overrightarrow{\psi }}_l\left(0\right)}{d{t}^2}}& =E^2{\displaystyle \frac{d^4{\overrightarrow{\psi }}_b\left(0\right)}{d{x}^4}}-2EB(0,0){\displaystyle \frac{d^2{\overrightarrow{\psi }}_b\left(0\right)}{d{x}^2}}+\left(B^2(0,0)-{\displaystyle \frac{\partial B(0,0)}{\partial t}}-E{\displaystyle \frac{{\partial }^{2}B(0,0)}{{\partial }^{2}x}}\right){\overrightarrow{\psi }}_b\left(0\right)\hfill \\ \hfill & -B(0,0)\overrightarrow{f}(0,0)+{\displaystyle \frac{\partial \overrightarrow{f}(0,0)}{\partial t}}+E{\displaystyle \frac{{\partial }^2\overrightarrow{f}(0,0)}{\partial x^2}},\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{d^2{\overrightarrow{\psi }}_r\left(0\right)}{d{t}^2}}& =E^2{\displaystyle \frac{d^4{\overrightarrow{\psi }}_b\left(1\right)}{d{x}^4}}-2EB(1,0){\displaystyle \frac{d^2{\overrightarrow{\psi }}_b\left(1\right)}{d{x}^2}}+\left(B^2(1,0)-{\displaystyle \frac{\partial B(1,0)}{\partial t}}-E{\displaystyle \frac{{\partial }^{2}B(1,0)}{{\partial }^{2}x}}\right){\overrightarrow{\psi }}_b\left(1\right)\hfill \\ \hfill & -B(1,0)\overrightarrow{f}(1,0)+{\displaystyle \frac{\partial \overrightarrow{f}(1,0)}{\partial t}}+E{\displaystyle \frac{{\partial }^2\overrightarrow{f}(1,0)}{\partial x^2}}.\hfill \end{array}$$

(10)

Then, there exists a solution $\overrightarrow{y}$ of (1) and (2) such that $\overrightarrow{y}\in {\left(C_{\lambda }^4\left(\overline{\mathsf{\Omega }}\right)\right)}^n$ where $C_{\lambda }^4\left(\overline{\mathsf{\Omega }}\right)$ is the space of Hölder continuous functions with $0<\lambda \le 1$.

Lemma 1 (Maximum Principle).

Let us assume that condition (4) holds and $\overrightarrow{\varphi }$ is any function in the domain of ${\overrightarrow{L}}_{\epsilon }$ such that $\overrightarrow{\varphi }\ge \overrightarrow{0}$ on Γ and ${\overrightarrow{L}}_{\epsilon }\overrightarrow{\varphi }\ge \overrightarrow{0}$ on Ω. Then, it is $\overrightarrow{\varphi }\ge \overrightarrow{0}$ on $\overline{\mathsf{\Omega }}.$

Proof. 

Let $i^{★},x^{★},t^{★}$ be such that ${\displaystyle {\varphi }_{i^{★}}\left(x^{★},t^{★}\right)=\underset{i,x,t}{\min }{\varphi }_i\left(x,t\right)}$. The result follows immediately if ${\varphi }_{i^{★}}\left(x^{★},t^{★}\right)\ge 0.$ Suppose ${\varphi }_{i^{★}}\left(x^{★},t^{★}\right)<0.$ Then, $\left(x^{★},t^{★}\right)\notin \mathsf{\Gamma },$ $\frac{\partial {\varphi }_{i^{★}}\left(x^{★},t^{★}\right)}{\partial t}=0$ and $\frac{{\partial }^2{\varphi }_{i^{★}}\left(x^{★},t^{★}\right)}{\partial x^2}\ge 0.$ Let $\left(x^{★},t^{★}\right)\in \mathsf{\Omega }.$ We have that

$${\left({\overrightarrow{L}}_{\epsilon }\overrightarrow{\varphi }\right)}_{i^{★}}\left(x^{★},t^{★}\right)\le \left\{\begin{array}{c}{\displaystyle \frac{\partial {\varphi }_{i^{★}}\left(x^{★},t^{★}\right)}{\partial t}-\epsilon \frac{{\partial }^2{\varphi }_{i^{★}}\left(x^{★},t^{★}\right)}{\partial x^2}+\sum _{j=1}^n{b}_{i^{★}j}\left(x^{★},t^{★}\right){\varphi }_{i^{★}}\left(x^{★},t^{★}\right),\mathrm{if}{i}^{★}=\left\{1,\dots ,m\right\},}\hfill \\ {\displaystyle \frac{\partial {\varphi }_{i^{★}}\left(x^{★},t^{★}\right)}{\partial t}-\frac{{\partial }^2{\varphi }_{i^{★}}\left(x^{★},t^{★}\right)}{\partial x^2}+\sum _{j=1}^n{b}_{i^{★}j}\left(x^{★},t^{★}\right){\varphi }_{i^{★}}\left(x^{★},t^{★}\right),\mathrm{if}{i}^{★}=\left\{m+1,\dots ,n\right\}.}\hfill \end{array}\right.$$

Using (4), we obtain ${\left({\overrightarrow{L}}_{\epsilon }\overrightarrow{\varphi }\right)}_{i^{★}}\left(x^{★},t^{★}\right)<0$, which is a contradiction. Thus, ${\varphi }_{i^{★}}\left(x^{★},t^{★}\right)\ge 0$, and the lemma is proven. □

Lemma 2 (Stability Result).

Let us assume that condition (4) holds and $\overrightarrow{\varphi }$ is any function in the domain of ${\overrightarrow{L}}_{\epsilon }.$ We find that for $(x,t)\in \overline{\mathsf{\Omega }}$ and $i=1,\dots ,n$, it holds that

$$\left|{\varphi }_i(x,t)\right|\le {∥\overrightarrow{\varphi }∥}_{\mathsf{\Gamma }}+\frac{1}{\alpha }∥{\overrightarrow{L}}_{\epsilon }\overrightarrow{\varphi }∥.$$

Proof. 

Let $A={∥\overrightarrow{\varphi }∥}_{\mathsf{\Gamma }}+\frac{1}{\alpha }∥{\overrightarrow{L}}_{\epsilon }\overrightarrow{\varphi }∥.$ Define ${\overrightarrow{\theta }}^{\pm }(x,t)=$ $A\overrightarrow{e}$ $\pm \overrightarrow{\varphi }(x,t)$, where $\overrightarrow{e}={(1,\dots ,1)}^T$. Then, ${\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on $\mathsf{\Gamma }$ and using (4), ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on $\mathsf{\Omega }.$ Hence, from Lemma 1, ${\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on $\overline{\mathsf{\Omega }},$ which proves the result. □

Lemma 3.

Let us assume that conditions (3), (4), (6)–(10) hold. In this case, for $(x,t)\in \overline{\mathsf{\Omega }}$, $i=1,\dots ,m$, $p=1,\dots ,n$, and $j=m+1,\dots ,n$, we get

$$\begin{array}{c}\left|y_p(x,t)\right|\le {∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+\frac{1}{\alpha }∥\overrightarrow{f}∥,\left|\frac{{\partial }^k{y}_p(x,t)}{\partial t^k}\right|\le C\left({\displaystyle {∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+\sum _{l=0}^k∥\frac{{\partial }^l\overrightarrow{f}}{\partial t^l}∥}\right),k=1,2,\hfill \\ \left|\frac{{\partial }^k{y}_i(x,t)}{\partial x^k}\right|\le C{\epsilon }^{-\frac{k}{2}}\left({∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+∥\overrightarrow{f}∥+∥\frac{\partial \overrightarrow{f}}{\partial t}∥\right),k=1,2,\hfill \\ \left|\frac{{\partial }^k{y}_j(x,t)}{\partial x^k}\right|\le C\left({∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+∥\overrightarrow{f}∥+∥\frac{\partial \overrightarrow{f}}{\partial t}∥\right),k=1,2,\hfill \\ \left|\frac{{\partial }^k{y}_i(x,t)}{\partial x^k}\right|\le C{\epsilon }^{-\frac{k}{2}}\left({∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+∥\overrightarrow{f}∥+\left.∥\frac{\partial \overrightarrow{f}}{\partial t}\right|+∥\frac{{\partial }^2\overrightarrow{f}}{\partial t^2}∥+{\epsilon }^{\frac{k-2}{2}}∥\frac{{\partial }^{k-2}\overrightarrow{f}}{\partial x^{k-2}}∥\right),k=3,4,\hfill \\ \left|\frac{{\partial }^k{y}_j(x,t)}{\partial x^k}\right|\le C{\epsilon }^{-\frac{(k-2)}{2}}\left({∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+∥\overrightarrow{f}∥+∥\frac{\partial \overrightarrow{f}}{\partial t}∥+∥\frac{{\partial }^2\overrightarrow{f}}{\partial t^2}∥+{\epsilon }^{\frac{k-2}{2}}∥\frac{{\partial }^{k-2}\overrightarrow{f}}{\partial x^{k-2}}∥\right),k=3,4,\hfill \\ \left|\frac{{\partial }^k{y}_i(x,t)}{\partial x^{k-1}\partial t}\right|\le C{\epsilon }^{\frac{1-k}{2}}\left({∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+∥\overrightarrow{f}∥+∥\frac{\partial \overrightarrow{f}}{\partial t}∥+∥\frac{{\partial }^2\overrightarrow{f}}{\partial t^2}∥\right),k=3,4,\hfill \\ \left|\frac{{\partial }^k{y}_j(x,t)}{\partial x^{k-1}\partial t}\right|\le C\left({∥\overrightarrow{y}∥}_{\mathsf{\Gamma }}+∥\overrightarrow{f}∥+∥\frac{\partial \overrightarrow{f}}{\partial t}∥+∥\frac{{\partial }^2\overrightarrow{f}}{\partial t^2}∥\right),k=2,3.\hfill \end{array}$$

Proof. 

The bound on $\overrightarrow{y}$ follows from Lemma 2. Differentiating the $p^{\mathrm{th}}$ equation in (1) partially with respect to t once and rearranging the terms, we obtain ${\displaystyle {\left({\overrightarrow{L}}_{\epsilon }\frac{\partial \overrightarrow{y}(x,t)}{\partial t}\right)}_p=\frac{\partial f_p(x,t)}{\partial t}-\sum _{l=1}^n\frac{\partial b_{pl}(x,t)}{\partial t}{y}_l(x,t).}$ Then, the bound on $\frac{\partial y_p(x,t)}{\partial t}$ follows by using Lemma 2. The bound on $\frac{{\partial }^2{y}_p(x,t)}{\partial t^2}$ can be derived in the same way. For each $(x,t),$ consider a neighborhood $I_x=[a,a+\sqrt{\epsilon }]$ such that $x\in I_x.$ Then, by using the mean value theorem with $\frac{\partial y_i}{\partial x}$, we find that for $z\in I_x$, it is

$$\frac{\partial y_i(z,t)}{\partial x}={\displaystyle \frac{y_i(a+\sqrt{\epsilon },t)-y_i(a,t)}{\sqrt{\epsilon }}},$$

and thus,

$$\left|\frac{\partial y_i(z,t)}{\partial x}\right|\le 2{\epsilon }^{-1/2}{∥y_i∥}_{I_x}.$$

(11)

Now, consider the relation

$$\frac{\partial y_i(x,t)}{\partial x}=\frac{\partial y_i(z,t)}{\partial x}+{\int }_z^x\frac{{\partial }^2{y}_i(s,t)}{\partial x^2}ds$$

(12)

$$\begin{array}{cc}\hfill =& {\displaystyle \frac{\partial y_i(z,t)}{\partial x}+{\epsilon }^{-1}{\int }_z^x\left(\frac{\partial y_i(s,t)}{\partial t}-f_i(s,t)+\sum _{l=1}^n\left(b_{il}{y}_l\right)(s,t)\right)ds.}\hfill \end{array}$$

(13)

Using the bounds of $\frac{\partial y_i}{\partial x},$ $\frac{\partial y_i}{\partial t},$ $f_i,b_{il}$ and $y_l,$ the bound of $\frac{\partial y_i(x,t)}{\partial x}$ follows from (12). The bound on $\frac{\partial y_j(x,t)}{\partial x}$ follows by using a similar procedure considering the interval $[a,a+\lambda ]$ where $0\le \lambda \le 1-a.$

Further, the bounds of the mixed derivatives can also be derived by using this same technique. From (1), we obtain

$$\frac{{\partial }^2\overrightarrow{y}(x,t)}{\partial x^2}=E^{-1}\left(\frac{\partial \overrightarrow{y}(x,t)}{\partial t}+B(x,t)\overrightarrow{y}(x,t)-\overrightarrow{f}(x,t)\right).$$

(14)

The bounds on $\frac{{\partial }^2\overrightarrow{y}(x,t)}{\partial x^2}$ follow from (14). Differentiating (14) with respect to x once and using the bounds of $\frac{{\partial }^2\overrightarrow{y}}{\partial x\partial t},$ $\frac{\partial \overrightarrow{y}}{\partial x},$ $\overrightarrow{y},$ $\frac{\partial B}{\partial x},$ B and $\frac{\partial \overrightarrow{f}}{\partial x},$ the bound on $\frac{{\partial }^3\overrightarrow{y}(x,t)}{\partial x^3}$ follow. In addition, differentiating (14) with respect to x twice and using the bounds of $\frac{{\partial }^3\overrightarrow{y}}{\partial x^2\partial t},$ $\frac{{\partial }^2\overrightarrow{y}}{\partial x^2},$ $\frac{\partial \overrightarrow{y}}{\partial x},$ $\overrightarrow{y},$ $\frac{{\partial }^{2}B}{\partial x^2}$, $\frac{\partial B}{\partial x}$, $B,$ $\frac{{\partial }^2\overrightarrow{f}}{\partial x^2}$ and $\frac{\partial \overrightarrow{f}}{\partial x},$ the bound on $\frac{{\partial }^4\overrightarrow{y}(x,t)}{\partial x^4}$ follows. □

Now, we decompose $\overrightarrow{y}$ into two parts $\overrightarrow{v}$ and $\overrightarrow{w}$ such that $\overrightarrow{y}=\overrightarrow{v}+\overrightarrow{w}$ with $\overrightarrow{v}={\overrightarrow{y}}_0+E{\overrightarrow{v}}_1$, where ${\overrightarrow{y}}_0$ is the solution of the reduced problem (5), ${\overrightarrow{v}}_1$ is the solution of

$${\overrightarrow{L}}_{\epsilon }{\overrightarrow{v}}_1(x,t)=\frac{{\partial }^2{\overrightarrow{y}}_0(x,t)}{\partial x^2}\mathrm{on}\mathsf{\Omega },{\overrightarrow{v}}_1(x,t)=\overrightarrow{0},\mathrm{on}\mathsf{\Gamma },$$

(15)

and the component $\overrightarrow{w}$ is the solution of

$${\overrightarrow{L}}_{\epsilon }\overrightarrow{w}(x,t)=\overrightarrow{0}\mathrm{on}\mathsf{\Omega },\overrightarrow{w}(x,t)=\overrightarrow{y}(x,t)-\overrightarrow{v}(x,t)\mathrm{on}\mathsf{\Gamma }.$$

(16)

We can decompose the component $\overrightarrow{w}$ further as ${\overrightarrow{w}}^l(x,t)$ and ${\overrightarrow{w}}^r(x,t)$ such that $\overrightarrow{w}(x,t)={\overrightarrow{w}}^l(x,t)+{\overrightarrow{w}}^r(x,t)$, where ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{w}}^l(x,t)=\overrightarrow{0}$ on $\mathsf{\Omega },$ ${\overrightarrow{w}}^l=\overrightarrow{w}$ on ${\mathsf{\Gamma }}_l,$ ${\overrightarrow{w}}^l=\overrightarrow{0}$ on ${\mathsf{\Gamma }}_b\cup {\mathsf{\Gamma }}_r$ and ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{w}}^r(x,t)=\overrightarrow{0}$ on $\mathsf{\Omega },$ ${\overrightarrow{w}}^r=\overrightarrow{w}$ on ${\mathsf{\Gamma }}_r,$ ${\overrightarrow{w}}^r=\overrightarrow{0}$ on ${\mathsf{\Gamma }}_b\cup {\mathsf{\Gamma }}_l$.

Lemma 4.

Let us assume that conditions (3), (4), (6)–(10) hold. In this case, for $(x,t)\in \overline{\mathsf{\Omega }}$, $i=1,\dots ,m$, $p=1,\dots ,n,$ and $j=m+1,\dots ,n$, we have

$$\begin{array}{cc}\hfill & \left|v_p(x,t)\right|\le C,\left|\frac{{\partial }^k{v}_p(x,t)}{\partial t^k}\right|\le C,\left|\frac{{\partial }^k{v}_p(x,t)}{\partial x^k}\right|\le C,k=1,2,\hfill \\ \hfill & \left|\frac{{\partial }^k{v}_i(x,t)}{\partial x^k}\right|\le C(1+{\epsilon }^{1-\frac{k}{2}}),\left|\frac{{\partial }^k{v}_j(x,t)}{\partial x^k}\right|\le C,k=3,4,\hfill \\ \hfill & \left|\frac{{\partial }^k{v}_p(x,t)}{\partial x^{k-1}\partial t}\right|\le C,k=2,3.\hfill \end{array}$$

Proof. 

From (5), it is not hard to see that for $(x,t)\in \overline{\mathsf{\Omega }}$ and $p=1,\dots ,n$, it holds that

$$\begin{array}{c}\left|\frac{{\partial }^k{y}_{0p}(x,t)}{\partial t^k}\right|\le C,k=0,1,2,\left|\frac{{\partial }^k{y}_{0p}(x,t)}{\partial x^k}\right|\le C,k=1,2,3,4,\mathrm{and}\left|\frac{{\partial }^k{y}_{0p}(x,t)}{\partial x^{k-1}\partial t}\right|\le C,k=2,3.\hfill \end{array}$$

From (15) and the above bounds, it is clear that ${\overrightarrow{v}}_1$ is the solution of a problem similar to (1) and (2). Hence, the estimates in Lemma 3 hold for the components of ${\overrightarrow{v}}_1$ and their derivatives after replacing $\overrightarrow{f}$ by $\frac{{\partial }^2{\overrightarrow{y}}_0(x,t)}{\partial x^2}$ and appropriate derivatives. Thus, the bounds of $\overrightarrow{v}$ and its derivatives follow from those of ${\overrightarrow{y}}_0,$ ${\overrightarrow{v}}_1$ and their derivatives. □

The layer functions $B^l,B^r,B^{★}$ associated with the solution $\overrightarrow{y}$ of (1) and (2) are defined for all $0\le x\le 1$ by

$$\begin{array}{c}{B}^l\left(x\right)=e^{-x\sqrt{\alpha }/\sqrt{\epsilon }},B^r\left(x\right)=e^{-(1-x)\sqrt{\alpha }/\sqrt{\epsilon }},B^{★}\left(x\right)=B^l\left(x\right)+B^r\left(x\right).\hfill \end{array}$$

Lemma 5.

Let us assume that conditions (3), (4), (6)–(10) hold. In this case, for $(x,t)\in \overline{\mathsf{\Omega }},$ $i=1,\dots ,m$, $j=m+1,\dots ,n$, we have that

$$\begin{array}{cc}\hfill & \left|\frac{{\partial }^k{w}_i^l(x,t)}{\partial t^k}\right|\le C_1{B}^l\left(x\right),\left|\frac{{\partial }^k{w}_j^l(x,t)}{\partial t^k}\right|\le C_2\epsilon \left(1-B^l\left(x\right)\right),k=0,1,2,\hfill \\ \hfill & \left|\frac{\partial w_i^l(x,t)}{\partial x}\right|\le C_1{\epsilon }^{-1/2}{B}^l\left(x\right)+C_2{\epsilon }^{1/2},\left|\frac{\partial w_j^l(x,t)}{\partial x}\right|\le C{\epsilon }^{1/2},\hfill \\ \hfill & \left|\frac{{\partial }^2{w}_i^l(x,t)}{\partial x^2}\right|\le C_1{\epsilon }^{-1}{B}^l\left(x\right)+C_2\left(1-B^l\left(x\right)\right),\left|\frac{{\partial }^2{w}_j^l(x,t)}{\partial x^2}\right|\le C_1{B}^l\left(x\right)+C_2\epsilon \left(1-B^l\left(x\right)\right),\hfill \\ \hfill & \left|\frac{{\partial }^3{w}_i^l(x,t)}{\partial x^3}\right|\le C{\epsilon }^{-1/2}\left(1+{\epsilon }^{-1}{B}^l\left(x\right)\right),\left|\frac{{\partial }^3{w}_j^l(x,t)}{\partial x^3}\right|\le C{\epsilon }^{1/2}\left(1+{\epsilon }^{-1}{B}^l\left(x\right)\right),\hfill \\ \hfill & \left|\frac{{\partial }^4{w}_i^l(x,t)}{\partial x^4}\right|\le C{\epsilon }^{-1}\left(1+{\epsilon }^{-1}{B}^l\left(x\right)\right),\left|\frac{{\partial }^4{w}_j^l(x,t)}{\partial x^4}\right|\le C\left(1+{\epsilon }^{-1}{B}^l\left(x\right)\right).\hfill \end{array}$$

Analogous results hold for $w_i^r,w_j^r$ and their derivatives with x replaced by $1-x.$

Proof. 

Define the functions ${\theta }_i^{\pm }(x,t)=C_1{B}^l\left(x\right)\pm w_i^l(x,t)$, $i=1,\dots ,m$ and ${\theta }_j^{\pm }(x,t)=C_2\epsilon (1-B^l\left(x\right))\pm w_j^l(x,t)$,$j=m+1,\dots ,n$. Then, ${\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on $\mathsf{\Gamma }$ for proper choices of $C_1$ and $C_2.$ Now, for $(x,t)\in \mathsf{\Omega }$, we get

$${\left({\overrightarrow{L}}_{\epsilon }{\overrightarrow{\theta }}^{\pm }\right)}_k(x,t)=\left\{\begin{array}{cc}{C}_1\left({\displaystyle \sum _{q=1}^n{b}_{kq}(x,t)-\alpha }\right)B^l\left(x\right),\hfill & \mathrm{if}k=1,\dots ,m,\hfill \\ {\displaystyle C_2\alpha B^l\left(x\right)+C_2\epsilon \left(1-B^l\left(x\right)\right)\sum _{q=1}^n{b}_{kq}(x,t),}\hfill & \mathrm{if}k=m+1,\dots ,n.\hfill \end{array}\right.$$

Using (4), we find that ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{\theta }}^{\pm }\ge \overrightarrow{0}$ on $\mathsf{\Omega }.$ By using Lemma 1 with ${\overrightarrow{\theta }}^{\pm }$, the bounds of $w_i^l$ and $w_j^l$ follow. Now, we derive the bounds on $\frac{\partial w_i^l}{\partial t}$ and $\frac{\partial w_j^l}{\partial t}.$ Define the functions ${\chi }_i^{\pm }(x,t)=C_1{B}^l\left(x\right)\pm \frac{\partial w_i^l(x,t)}{\partial t}$, $i=1,\dots ,m$ and ${\chi }_j^{\pm }(x,t)=C_2\epsilon \left(1-B^l\left(x\right)\right)\pm \frac{\partial w_j^l(x,t)}{\partial t}$, $j=m+1,\dots ,n$. Differentiate the equation ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{w}}^l=\overrightarrow{0}$ partially with respect to t once and rearrange the terms to get

$${\left({\overrightarrow{L}}_{\epsilon }\frac{\partial {\overrightarrow{w}}^l}{\partial t}\right)}_k=\left\{\begin{array}{cc}{\displaystyle \frac{{\partial }^2{w}_k^l}{\partial t^2}-\epsilon \frac{{\partial }^3{w}_k^l}{\partial x^2\partial t}+\sum _{p=1}^n{b}_{kp}\frac{\partial w_p^l}{\partial t}=-\sum _{p=1}^n\frac{\partial b_{kp}}{\partial t}{w}_p^l,}\hfill & \mathrm{if}k=1,\dots ,m,\hfill \\ {\displaystyle \frac{{\partial }^2{w}_k^l}{\partial t^2}-\frac{{\partial }^3{w}_k^l}{\partial x^2\partial t}+\sum _{p=1}^n{b}_{kp}\frac{\partial w_p^l}{\partial t}=-\sum _{p=1}^n\frac{\partial b_{kp}}{\partial t}{w}_p^l,}\hfill & \mathrm{if}k=m+1,\dots ,n.\hfill \end{array}\right.$$

(17)

Hence,

$$\left|{\left({\overrightarrow{L}}_{\epsilon }\frac{\partial {\overrightarrow{w}}^l(x,t)}{\partial t}\right)}_k\right|\le \left\{\begin{array}{cc}{C}_1{B}^l\left(x\right),\hfill & \mathrm{if}k=1,\dots ,m,\hfill \\ C_2\epsilon \left(1-B^l\left(x\right)\right),\hfill & \mathrm{if}k=m+1,\dots ,n.\hfill \end{array}\right.$$

(18)

Using Lemmas 3 and 4, we obtain that ${\left|\frac{\partial w_k^l(x,t)}{\partial t}\right|}_{\mathsf{\Gamma }}\le C$. Thus, the bounds of $\frac{\partial w_i^l}{\partial t}$ and $\frac{\partial w_j^l}{\partial t}$ follow from using Lemma 1 applied to the functions ${\chi }_k^{\pm }$. From Lemmas 3 and 4, we get

$$\begin{array}{c}\left|\frac{{\partial }^2{w}_i^l(x,t)}{\partial x\partial t}\right|\le C{\epsilon }^{-1/2},\left|\frac{{\partial }^2{w}_j^l(x,t)}{\partial x\partial t}\right|\le C,\left|\frac{{\partial }^3{w}_i^l(x,t)}{\partial x^2\partial t}\right|\le C{\epsilon }^{-1},\left|\frac{{\partial }^3{w}_j^l(x,t)}{\partial x^2\partial t}\right|\le C.\hfill \end{array}$$

(19)

It is not hard to verify that $\left|\frac{{\partial }^{k+p}{w}_i^l(x,t)}{\partial x^k\partial t^p}\right|\le C{\epsilon }^{-k/2}{B}^l\left(x\right)$ and $\left|\frac{{\partial }^{k+p}{w}_j^l(x,t)}{\partial x^k\partial t^p}\right|$≤$C{B}^l\left(x\right),$ $k\le 3,p\le 2$ and $0\le k+2p\le 4.$ Using the above estimates in (17), it can be obtained that $\left|\frac{{\partial }^2{w}_p^l(x,t)}{\partial t^2}\right|\le C.$ Now, define the functions ${\mathsf{{\rm Y}}}_i^{\pm }(x,t)=C_1{e}^{\alpha t}{B}^l\left(x\right)\pm \frac{{\partial }^2{w}_i^l(x,t)}{\partial t^2}$, $i=1,\dots ,m$ and ${\mathsf{{\rm Y}}}_j^{\pm }(x,t)=C_2\epsilon e^{\alpha t}\left(1-B^l\left(x\right)\right)\pm \frac{{\partial }^2{w}_j^l(x,t)}{\partial t^2}$, $j=m+1,\dots ,n$. Using Lemma 1 with the functions ${\mathsf{{\rm Y}}}_q^{\pm }$, the bounds of $\frac{{\partial }^2{w}_i^l}{\partial t^2}$ and $\frac{{\partial }^2{w}_j^l}{\partial t^2}$ follow. Using the mean value theorem with $I_x$ and similar arguments to those used to get a bound of $\frac{\partial y_i(x,t)}{\partial x}$, we deduce that

$$\left|\frac{\partial w_i^l(x,t)}{\partial x}\right|\le C{\epsilon }^{-1/2}\left(B^l\left(x\right)+\sum _{k=1}^n{∥w_k^l∥}_{I_x}\right).$$

(20)

Since ${\left|w_i^l\right|}_{I_x}\le C{B}^l\left(x\right)$ and ${∥w_j^l∥}_{I_x}\le C\epsilon ,$ the bounds on $\frac{\partial w_i^l(x,t)}{\partial x}$ follow from (20). Using a similar procedure, the bounds on $\frac{\partial w_j^l(x,t)}{\partial x}$ can be derived. The bounds on $\frac{{\partial }^2{w}_q^l(x,t)}{\partial x^2},$ $q=1,\dots ,n$ follow from the equation ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{w}}^l=\overrightarrow{0}.$ Differentiating the equation ${\overrightarrow{L}}_{\epsilon }{\overrightarrow{w}}^l=\overrightarrow{0}$ once and twice partially with respect to x and using the above bounds, the bounds on $\frac{{\partial }^3{w}_q^l(x,t)}{\partial x^3}$ and $\frac{{\partial }^4{w}_q^l(x,t)}{\partial x^4},$ $q=1,\dots ,n$ can be readily obtained. □

4. Discretization of the Domain and Discretized Problem

In this section, a finite difference scheme using a piecewise uniform Shishkin mesh with a well-suited transition parameter for the SPPDE (1) and (2) is considered (see [1] and references therein). A rectangular mesh ${\overline{\mathsf{\Omega }}}^{M,N}$ with $M\times N$ mesh regions is now constructed on the domain $\overline{\mathsf{\Omega }}.$ The rectangular mesh ${\overline{\mathsf{\Omega }}}^{M,N}$ consists of a piecewise uniform Shishkin mesh on the spatial domain $[0,1]$ and a uniform mesh on the temporal domain $[0,T]$. Let ${\mathsf{\Omega }}_x^N=$ ${\left\{x_j\right\}}_{j=1}^{N-1}$, ${\overline{\mathsf{\Omega }}}_x^N=$ ${\left\{x_j\right\}}_{j=0}^N$, ${\mathsf{\Omega }}_t^M=$ ${\left\{t_k\right\}}_{k=1}^M$, ${\overline{\mathsf{\Omega }}}_t^M=$ ${\left\{t_k\right\}}_{k=0}^M$, ${\mathsf{\Omega }}^{M,N}=$ ${\mathsf{\Omega }}_t^M\times {\mathsf{\Omega }}_x^N$, ${\overline{\mathsf{\Omega }}}^{M,N}=$ ${\overline{\mathsf{\Omega }}}_t^M\times {\overline{\mathsf{\Omega }}}_x^N$ and ${\mathsf{\Gamma }}^{M,N}=$ ${\overline{\mathsf{\Omega }}}^{M,N}\cap \mathsf{\Gamma }$. On the spatial domain $[0,1]$, a piecewise uniform Shishkin mesh ${\overline{\mathsf{\Omega }}}_x^N$ with N mesh intervals is constructed using a transition parameter $\eta$ as follows. Thus, we split the interval [0,1] as

$$[0,\eta ]\cup (\eta ,1-\eta ]\cup (1-\eta ,1],$$

where the parameter $\eta$ is a function of $\alpha$, $\epsilon$ and N, given by

$$\eta =\min \left\{\frac{1}{4},\frac{2\sqrt{\epsilon }}{\sqrt{\alpha }}\ln N\right\}.$$

Each of the sub-intervals $[0,\eta ]$ and $(1-\eta ,1]$ is divided into $\frac{N}{4}$ subintervals using appropriate uniform mesh points, whereas $(\eta ,1-\eta ]$ is divided into $\frac{N}{2}$ sub-intervals using also appropriate uniform mesh points. On the temporal domain $[0,T]$ a uniform mesh ${\overline{\mathsf{\Omega }}}_t^M$ with M mesh intervals is placed. The discrete problem corresponding to (1) and (2) is

$$\begin{array}{c}{D}_t^{-}\overrightarrow{Y}(x_j,t_k)-E{\delta }_x^2\overrightarrow{Y}(x_j,t_k)+B(x_j,t_k)\overrightarrow{Y}(x_j,t_k)=\overrightarrow{f}(x_j,t_k)\mathrm{on}{\mathsf{\Omega }}^{M\times N}\mathrm{with}\overrightarrow{Y}=\overrightarrow{y}\mathrm{on}{\mathsf{\Gamma }}^{M,N}.\hfill \end{array}$$

(21)

Here,

$D_t^{-}\mathsf{\Psi }(x_j,t_k)$ $={\displaystyle \frac{\mathsf{\Psi }(x_j,t_k)-\mathsf{\Psi }(x_j,t_{k-1})}{h_t}},$ $h_t=t_k-t_{k-1},$ ${\delta }_x^2\mathsf{\Psi }(x_j,t_k)=$ ${\displaystyle \frac{(D_x^{+}-D_x^{-})\mathsf{\Psi }(x_j,t_k)}{{\overline{h}}_j}},$ $D_x^{-}\mathsf{\Psi }(x_j,t_k)$ $={\displaystyle \frac{\mathsf{\Psi }(x_j,t_k)-\mathsf{\Psi }(x_{j-1},t_k)}{h_j}},$ $D_x^{+}\mathsf{\Psi }(x_j,t_k)$ $={\displaystyle \frac{\mathsf{\Psi }(x_{j+1},t_k)-\mathsf{\Psi }(x_j,t_k)}{h_{j+1}}},$ $h_j=x_j-x_{j-1}$ and ${\overline{h}}_j={\displaystyle \frac{h_{j+1}+h_j}{2}}$ such that ${\overline{h}}_0={\displaystyle \frac{h_1}{2}}$ and ${\overline{h}}_N={\displaystyle \frac{h_N}{2}}.$

In operator form, the problem (21) is written as follows:

$${\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{Y}\left(x_j,t_k\right)=\overrightarrow{f}\left(x_j,t_k\right),\left(x_j,t_k\right)\in {\mathsf{\Omega }}^{M,N}\mathrm{with}\overrightarrow{Y}=\overrightarrow{y}\mathrm{on}{\mathsf{\Gamma }}^{M,N}.$$

(22)

Thus, the operator ${\overrightarrow{L}}_{\epsilon }^{M,N}$ is defined by ${\overrightarrow{L}}_{\epsilon }^{M,N}=I{D}_t^{-}-E{\delta }_x^2+B.$

Lemma 6.

[Discrete Maximum Principle] Let us assume that condition (4) holds and $\overrightarrow{\mathsf{\Phi }}$ is any mesh function such that $\overrightarrow{\mathsf{\Phi }}\ge \overrightarrow{0}$ on ${\mathsf{\Gamma }}^{M,N}$ and, ${\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{\mathsf{\Phi }}\ge \overrightarrow{0}$ on ${\mathsf{\Omega }}^{M,N}$. Then, it holds that $\overrightarrow{\mathsf{\Phi }}\ge \overrightarrow{0}$ on ${\overline{\mathsf{\Omega }}}^{M,N}.$

Proof. 

Take $j^{★},i^{★},k^{★},$ for which ${\displaystyle {\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)=\underset{i,j,k}{\min }{\mathsf{\Phi }}_i\left(x_j,t_k\right)}$. The result follows immediately if ${\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)\ge 0.$ Suppose ${\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)<0.$ Then $\left(x_{j^{★}},t_{k^{★}}\right)$∉${\mathsf{\Gamma }}^{M,N}$, $D_t^{-}{\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)\le 0$ and ${\delta }_x^2{\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)\ge 0.$ Let $\left(x_{j^{★}},t_{k^{★}}\right)\in {\mathsf{\Omega }}^{M,N}.$ Now, we have

$${\left({\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{\mathsf{\Phi }}\right)}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)\le \left\{\begin{array}{c}{\displaystyle D_t^{-}{\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)-\epsilon {\delta }_x^2{\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)+\sum _{l=1}^n{b}_{i^{★}l}\left(x_{j^{★}},t_{k^{★}}\right){\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right),\mathrm{if}{i}^{★}=1,\dots ,m,}\hfill \\ {\displaystyle D_t^{-}{\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)-{\delta }_x^2{\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)+\sum _{l=1}^n{b}_{i^{★}l}\left(x_{j^{★}},t_{k^{★}}\right){\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right),\mathrm{if}{i}^{★}=m+1,\dots ,n.}\hfill \end{array}\right.$$

Using (4), we get ${\left({\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{\mathsf{\Phi }}\right)}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)<0,$ which is a contradiction. Thus, ${\mathsf{\Phi }}_{i^{★}}\left(x_{j^{★}},t_{k^{★}}\right)$≥ 0, and the lemma is proven. □

Lemma 7 (Discrete Stability Result).

Let us assume that condition (4) holds and $\overrightarrow{\mathsf{\Phi }}$ is any mesh function. Then, for $\left(x_j,t_k\right)\in {\overline{\mathsf{\Omega }}}^{M,N}$ and $i=1,\dots ,n$, we have that

$$\left|{\mathsf{\Phi }}_i\left(x_j,t_k\right)\right|\le {∥\overrightarrow{\mathsf{\Phi }}∥}_{{\mathsf{\Gamma }}^{M,N}}+\frac{1}{\alpha }∥{\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{\mathsf{\Phi }}∥.$$

Proof. 

Let $A={∥\overrightarrow{\mathsf{\Phi }}∥}_{{\mathsf{\Gamma }}^{M,N}}+\frac{1}{\alpha }∥{\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{\mathsf{\Phi }}∥.$ Define ${\overrightarrow{\mathsf{\Theta }}}^{\pm }\left(x_j,t_k\right)=$ $A\overrightarrow{e}$±$\overrightarrow{\mathsf{\Phi }}\left(x_j,t_k\right)$, where $\overrightarrow{e}={(1,\dots ,1)}^T.$ Then ${\overrightarrow{\mathsf{\Theta }}}^{\pm }\ge \overrightarrow{0}$ on ${\mathsf{\Gamma }}^{M,N}$ and using (4) we find that ${\overrightarrow{L}}_{\epsilon }^{M,N}{\overrightarrow{\mathsf{\Theta }}}^{\pm }$≥$\overrightarrow{0}$ on ${\mathsf{\Omega }}^{M,N}.$ Hence, from Lemma 6, we have ${\overrightarrow{\mathsf{\Theta }}}^{\pm }\ge \overrightarrow{0}$ on ${\overline{\mathsf{\Omega }}}^{M,N}$, which proves the result. □

5. Error Analysis

This section gives an error bound of the solution provided by the present method. We began by considering the decomposition of the solution of the discrete problem (21). This is similar to the continuous case. The solution $\overrightarrow{Y}$ of the problem (21) can be decomposed into $\overrightarrow{V}$ and $\overrightarrow{W}$ such that $\overrightarrow{Y}=\overrightarrow{V}+\overrightarrow{W}$, where $\overrightarrow{V}$ is the solution of

$$\begin{array}{c}{\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{V}(x_j,t_k)=\overrightarrow{f}(x_j,t_k),(x_j,t_k)\in {\mathsf{\Omega }}^{M,N}\mathrm{with}\overrightarrow{V}=\overrightarrow{v}\mathrm{on}{\mathsf{\Gamma }}^{M,N},\end{array}$$

(23)

and $\overrightarrow{W}$ is the solution of

$$\begin{array}{c}{\overrightarrow{L}}_{\epsilon }^{M,N}\overrightarrow{W}(x_j,t_k)=\overrightarrow{0},(x_j,t_k)\in {\mathsf{\Omega }}^{M,N}\mathrm{with}\overrightarrow{W}=\overrightarrow{w}\mathrm{on}{\mathsf{\Gamma }}^{M,N}.\end{array}$$

(24)

It is noted that for any smooth function $\varphi$, we have

$$\begin{array}{c}\hfill \left|\left(D_t^{-}-\frac{\partial }{\partial t}\right)\varphi (x_j,t_k)\right|\le C\left(t_k-t_{k-1}\right)\underset{s\in J_k}{\max }\left|\frac{{\partial }^2\varphi (x_j,s)}{\partial t^2}\right|,\end{array}$$

(25)

$$\begin{array}{c}\hfill \left|\left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)\varphi (x_j,t_k)\right|\le C{\left(x_{j+1}-x_{j-1}\right)}^2\underset{s\in I_j}{\max }\left|\frac{{\partial }^4\varphi (s,t_k)}{\partial x^4}\right|,\end{array}$$

(26)

and

$$\left|{\delta }_x^2\varphi \left(x_j,t_k\right)\right|\le \underset{s\in I_j}{\max }\left|\frac{{\partial }^2\varphi (s,t_k)}{\partial x^2}\right|,$$

(27)

where, $J_k=[t_{k-1},t_k]$ and $I_j=[x_{j-1},x_{j+1}].$

Theorem 2.

Let $\overrightarrow{v}$ and $\overrightarrow{w}$ be the components of $\overrightarrow{y}$, and let $\overrightarrow{V}$ and $\overrightarrow{W}$ be the components of $\overrightarrow{Y}$. Then, for all $(x_j,t_k)\in {\overline{\mathsf{\Omega }}}^{M,N}$ and $p=1,\dots ,n$, it holds that

$$\begin{array}{cc}\hfill \left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{V}-\overrightarrow{v}\right)\right)}_p\left(x_j,t_k\right)\right|& \le C\left(M^{-1}+N^{-2}\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-\overrightarrow{w}\right)\right)}_p\left(x_j,t_k\right)\right|& \le C\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right).\hfill \end{array}$$

(29)

Proof. 

From the defining equation for $\overrightarrow{v}$ and (23), we have

$${\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{V}-\overrightarrow{v}\right)\right)}_i=\left\{\begin{array}{cc}\left(D_t^{-}-\frac{\partial }{\partial t}\right)v_i-\epsilon \left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)v_i,\hfill & \mathrm{if}i=1,\dots ,m,\hfill \\ \left(D_t^{-}-\frac{\partial }{\partial t}\right)v_i-\left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)v_i,\hfill & \mathrm{if}i=m+1,\dots ,n,\hfill \end{array}\right.$$

(30)

and from (16) and (24), we have

$${\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-\overrightarrow{w}\right)\right)}_i=\left\{\begin{array}{c}\left(D_t^{-}-\frac{\partial }{\partial t}\right)w_i-\epsilon \left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)w_i,\mathrm{if}i=1,\dots ,m,\hfill \\ \left(D_t^{-}-\frac{\partial }{\partial t}\right)w_i-\left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)w_i,\mathrm{if}i=m+1,\dots ,n.\hfill \end{array}\right.$$

(31)

Note that for all $k,$ $t_k-t_{k-1}=\frac{T}{M}.$ Since $x_{j+1}-x_{j-1}\le 2N^{-1}$, for any choice of the parameter $\eta ,$ from (25), (26), and (30), for $i=1,\dots ,m$ and $q=m+1,\dots ,n$, we have

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{V}-\overrightarrow{v}\right)\right)}_i\left(x_j,t_k\right)\right|\le C\left(M^{-1}{\left|\frac{{\partial }^2{v}_i}{\partial t^2}\right|}_{J_k}+\epsilon N^{-2}{\left|\frac{{\partial }^4{v}_i}{\partial x^4}\right|}_{I_j}\right),$$

(32)

and

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{V}-\overrightarrow{v}\right)\right)}_q\left(x_j,t_k\right)\right|\le C\left(M^{-1}{\left|\frac{{\partial }^2{v}_q}{\partial t^2}\right|}_{J_k}+N^{-2}{\left|\frac{{\partial }^4{v}_q}{\partial x^4}\right|}_{I_j}\right).$$

(33)

Then, using Lemma 4, (28) follows from (32) and (33) for $p=1,\dots ,n$. Now, we estimate the error in ${\overrightarrow{w}}^l.$ From (31), we obtain

$${\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_i=\left\{\begin{array}{c}\left(D_t^{-}-\frac{\partial }{\partial t}\right)w_i^l-\epsilon \left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)w_i^l,\mathrm{if}i=1,\dots ,m,\hfill \\ \left(D_t^{-}-\frac{\partial }{\partial t}\right)w_i^l-\left({\delta }_x^2-\frac{{\partial }^2}{\partial x^2}\right)w_i^l,\mathrm{if}i=m+1,\dots ,n.\hfill \end{array}\right.$$

(34)

Here, we consider that the argument depends on whether $\eta =\frac{1}{4}$ or $\eta =\frac{2\sqrt{\epsilon }}{\sqrt{\alpha }}\ln N.$ If $\eta =\frac{1}{4}$, then ${\epsilon }^{-1}\le C{\left(\ln N\right)}^2$, and it follows $x_{j+1}-x_{j-1}=N^{-1}$ in this case. Using (25), (26), and (34), for $i=1,\dots ,m$ and $q=m+1,\dots ,n$, we get

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_i\left(x_j,t_k\right)\right|\le C\left(M^{-1}{\left|\frac{{\partial }^2{w}_i^l}{\partial t^2}\right|}_{J_k}+\epsilon N^{-2}{\left|\frac{{\partial }^4{w}_i^l}{\partial x^4}\right|}_{I_j}\right),$$

(35)

and

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_q\left(x_j,t_k\right)\right|\le C\left(M^{-1}{\left|\frac{{\partial }^2{w}_q^l}{\partial t^2}\right|}_{J_k}+N^{-2}{\left|\frac{{\partial }^4{w}_q^l}{\partial x^4}\right|}_{I_j}\right).$$

(36)

Later, using Lemma 5, for $p=1,\dots ,n$, we get

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_p\left(x_j,t_k\right)\right|\le C\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right).$$

Now, assume $\eta =\frac{2\sqrt{\epsilon }}{\sqrt{\alpha }}\ln N$. Let us prove the results on the domains $[0,\eta ]\times [0,T],$ $(1-\eta ,1]\times [0,T]$ and $(\eta ,1-\eta ]\times [0,T]$ separately. The spatial mesh spacing on both intervals $[0,\eta ]$ and $(1-\eta ,1]$ is $4N^{-1}\eta =\frac{8N^{-1}\sqrt{\epsilon }\ln N}{\sqrt{\alpha }}.$ Using (25), (26) and (34) for both $x_j\in [0,\eta ]$ and $x_j\in (1-\eta ,1]$ and Lemma 5, for $p=1,\dots ,n$, we get

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_p\left(x_j,t_k\right)\right|\le C\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right).$$

Finally, consider the interval $(\eta ,1-\eta ].$ Here, the spatial mesh spacing is $\frac{2(1-2\eta )}{N}$≤$C{N}^{-1}.$ From (25), (26), (34) and using Lemma 5, for $i=1,\dots ,m$ and $q=m+1,\dots ,n$, we have

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_i\left(x_j,t_k\right)\right|\le C\left(M^{-1}+\epsilon {\left(x_{j+1}-x_{j-1}\right)}^2{\left|\frac{\partial w_i^l}{\partial x^4}\right|}_{I_j}\right),$$

(37)

and

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_q\left(x_j,t_k\right)\right|\le C\left(M^{-1}+{\left(x_{j+1}-x_{j-1}\right)}^2{\left|\frac{\partial w_q^l}{\partial x^4}\right|}_{I_j}\right).$$

(38)

From (25), (27), (34) and using Lemma 5, for $i=1,\dots ,m$ and $q=m+1,\dots ,n$, we obtain

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_i\left(x_j,t_k\right)\right|\le C\left(M^{-1}+\epsilon {\left|\frac{{\partial }^2{w}_i^l}{\partial x^2}\right|}_{I_j}\right),$$

(39)

and

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_q\left(x_j,t_k\right)\right|\le C\left(M^{-1}+{\left|\frac{{\partial }^2{w}_q^l}{\partial x^2}\right|}_{I_j}\right).$$

(40)

It is not difficult to get

$$\underset{x\in I_j}{\max }\left\{B^l\left(x\right),B^r\left(x\right)\right\}\le C{N}^{-2},\eta <x_j\le 1-\eta .$$

(41)

Now, we proceed with the proof by considering two instances: $\epsilon \ge N^{-2}$ and $\epsilon \le N^{-2}$. In the first instance, using Lemma 5 and (41) in (37) and (38), for $p=1,\dots ,n$, we get

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_p(x_j,t_k)\right|\le C\left(M^{-1}+N^{-2}\right).$$

Consider the second choice $\epsilon \le N^{-2}.$ In this case, using Lemma 5 and (41) in (39) and (40), for $p=1,\dots ,n$, we have

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_p\left(x_j,t_k\right)\right|\le C\left(M^{-1}+N^{-2}\right).$$

Using similar arguments, we can estimate the error in ${\overrightarrow{w}}^r.$ By combining the error estimates for ${\overrightarrow{w}}^l$ and ${\overrightarrow{w}}^r,$ (29) holds in all cases for $p=1,\dots ,n$. □

Theorem 3.

Let us denote by $\overrightarrow{y}$ the solution of (1) and (2) and by $\overrightarrow{Y}$ the solution of (21). Then, for all $(x_j,t_k)\in {\overline{\mathsf{\Omega }}}^{M,N}$ and $p=1,\dots ,n$, we have

$$\left|{\left(\overrightarrow{Y}\left(x_j,t_k\right)-\overrightarrow{y}\left(x_j,t_k\right)\right)}_p\right|\le C\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right).$$

(42)

Proof. 

Note that for all $(x_j,t_k)\in {\overline{\mathsf{\Omega }}}^{M,N}$ and $p=1,\dots ,n$,

$$\begin{array}{c}\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{Y}-\overrightarrow{y}\right)\right)}_p\left(x_j,t_k\right)\right|\le \left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{V}-\overrightarrow{v}\right)\right)}_p\left(x_j,t_k\right)\right|+\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{W}-{\overrightarrow{w}}^l\right)\right)}_p\left(x_j,t_k\right)\right|.\hfill \end{array}$$

(43)

Using Theorem 2 with (43), one can obtain

$$\left|{\left({\overrightarrow{L}}_{\epsilon }^{M,N}\left(\overrightarrow{Y}-\overrightarrow{y}\right)\right)}_p\left(x_j,t_k\right)\right|\le C\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right).$$

(44)

Therefore, by using Lemma 7 with (44), for all $(x_j,t_k)\in {\overline{\mathsf{\Omega }}}^{M,N}$ and $p=1,\dots ,n$, we obtain the desired error bound, which is given by

$$\left|{\left(\overrightarrow{Y}\left(x_j,t_k\right)-\overrightarrow{y}\left(x_j,t_k\right)\right)}_p\right|\le C\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right).$$

This completes the proof. □

The obtained error bound provides the parameter uniform estimate as the constant C is independent of the perturbation parameter $\epsilon$. Furthermore, the numerical solution converges to the continuous solution almost linearly.

6. Numerical Illustrations

In this section, we consider a trial problem to investigate the effectiveness of the present method. The notations $D^{M,N},$ $C_p^{M,N}$, and $p^{M,N}$ correspond to the parameter-uniform maximum pointwise errors, the parameter-uniform error constants, and the parameter-uniform rates of convergence, respectively. Let $\overrightarrow{Y}(x_j,t_k)$ be an approximation of the true solution $\overrightarrow{y}(x_j,t_k)$ on the mesh $(x_j,t_k)\in {\mathsf{\Omega }}^{M,N}$, where M and N are the number of mesh points on each direction. We take $M\in R_M=\left\{40,80,160,320,640\right\}$ for the time-direction, and $N\in R_N=\left\{128,256,512,1024,2048\right\}$ for the space-direction. The exact solution for the considered test problem is not available, so for a finite set of values of the perturbation parameter $\epsilon \in R_{\epsilon }=\left\{2^{-3},\dots ,2^{-27}\right\}$, the parameter-uniform maximum errors are obtained as

$$\begin{array}{c}\hfill D^{M,N}=\underset{\epsilon }{\max }{D}_{\epsilon }^{M,N},\mathrm{where},D_{\epsilon }^{M,N}=\left\{\begin{array}{cc}\Vert {\overrightarrow{Y}}^{M,N}-{\overrightarrow{Y}}^{M,2N}\Vert ,\hfill & \mathrm{for}x,\hfill \\ \Vert {\overrightarrow{Y}}^{M,N}-{\overrightarrow{Y}}^{2M,N}\Vert ,\hfill & \mathrm{for}t.\hfill \end{array}\right.\end{array}$$

From these values, one can obtain the parameter-uniform rate of convergence using

$$\begin{array}{c}\hfill p^{M,N}=\left\{\begin{array}{cc}{\log }_2{\displaystyle \frac{D^{M,N}}{D^{M,2N}}},\hfill & \mathrm{for}x,\hfill \\ {\log }_2{\displaystyle \frac{D^{M,N}}{D^{2M,N}}},\hfill & \mathrm{for}t,\hfill \end{array}\right.\end{array}$$

and the parameter-uniform error constant given by

$$\begin{array}{c}\hfill C_p^{M,N}=\left\{\begin{array}{cc}\frac{D^{M,N}{N}^{p^{★}}}{1-2^{-p^{★}}},\hfill & \mathrm{for}x,\hfill \\ \frac{D^{M,N}{M}^{p^{★}}}{1-2^{-p^{★}}},\hfill & \mathrm{for}t,\hfill \end{array}\right.,\mathrm{where},p^{★}=\left\{\begin{array}{cc}{\displaystyle \underset{N}{\min }{p}^{M,N},}\hfill & \mathrm{for}x,\hfill \\ {\displaystyle \underset{M}{\min }{p}^{M,N},}\hfill & \mathrm{for}t.\hfill \end{array}\right.\end{array}$$

Example 1.

Consider the following system of parabolic equation for $(x,t)\in (0,1)\times (0,1],$

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial y_1(x,t)}{\partial t}}-\epsilon {\displaystyle \frac{{\partial }^2{y}_1(x,t)}{\partial x^2}}+2y_1(x,t)-t{y}_2(x,t)=t^3{e}^{-t}+{\left(xt\right)}^2,\hfill \\ \hfill & {\displaystyle \frac{\partial y_2(x,t)}{\partial t}}-{\displaystyle \frac{{\partial }^2{y}_2(x,t)}{\partial x^2}}-x{y}_1(x,t)+(2+t)y_2(x,t)=\sqrt{2}x{t}^2,\hfill \end{array}$$

with $\overrightarrow{y}(x,0)=\overrightarrow{0}$ on ${\mathsf{\Gamma }}_b,$ $\overrightarrow{y}(0,t)={(t^3-t^4,t^3)}^T$ on ${\mathsf{\Gamma }}_l$ and $\overrightarrow{y}(1,t)={(t^3,t^4)}^T$ on ${\mathsf{\Gamma }}_r.$

It is to be noted that the above problem satisfies conditions (3) and (4) and all the compatibility conditions (6)–(10). For different values of $\epsilon ,$ the values of $D^{M,N},$ $C_p^{M,N}$, and $p^{M,N}$ in the variable x with a uniform mesh consist of 10 mesh intervals for the variable t given in Table 1, and for different values of $\epsilon ,$ the values of $D^{M,N},$ $C_p^{M,N}$, and $p^{M,N}$ in the variable t with a piecewise uniform Shishkin mesh consist of 128 mesh intervals for the variable x given in Table 2. From Table 1 and Table 2, it can be observed that the errors are independent of the singular perturbation parameter $\epsilon$ and decrease as the numbers of mesh points M and N increase. Surface plots of the maximum error of the solution of the above test problem are presented in Figure 1 and Figure 2. For $M=10,$ $N=128$, and $\epsilon =2^{-15},$ Figure 1 shows the numerical solution of the component $y_1$ of the solution $\overrightarrow{y}={(y_1,y_2)}^T$. It is observed that the component $y_1$ changes rapidly near ${\mathsf{\Gamma }}_l$ and ${\mathsf{\Gamma }}_r$ (boundary layers) due to the presence of the perturbation parameter. Figure 2 shows the numerical solution of the component $y_2$, but it varies smoothly throughout the domain. Furthermore, Figure 3 shows the numerical solution of $\overrightarrow{y}.$ $Log-logplots$ for the errors given in Figure 4 and Figure 5, which clearly indicate that the error bound is $O\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right)$, as expected.

Table 1. Values of $D^{M,N},p^{M,N},$ $C_p^{M,N}$ for $M=10$ and $\alpha =0.9$ for Example 1.

$\mathit{\epsilon }$	Number of Mesh Points N
	128	256	512	1024	2048
$2^{-3}$	1.2939 × 10${}^{-5}$	3.2353 × 10${}^{-6}$	8.0886 × 10${}^{-7}$	2.0222 × 10${}^{-7}$	5.0576 × 10${}^{-8}$
$2^{-6}$	1.0664 × 10${}^{-4}$	2.6694 × 10${}^{-5}$	6.6757 × 10${}^{-6}$	1.6691 × 10${}^{-6}$	4.1729 × 10${}^{-7}$
$2^{-9}$	8.4547 × 10${}^{-4}$	2.1354 × 10${}^{-4}$	5.3581 × 10${}^{-5}$	1.3410 × 10${}^{-5}$	3.3532 × 10${}^{-6}$
$2^{-12}$	1.8819 × 10${}^{-3}$	9.7750 × 10${}^{-4}$	4.1598 × 10${}^{-4}$	1.6704 × 10${}^{-4}$	2.6836 × 10${}^{-5}$
$2^{-15}$	1.8813 × 10${}^{-3}$	9.7737 × 10${}^{-4}$	4.1593 × 10${}^{-4}$	1.7348 × 10${}^{-4}$	6.0595 × 10${}^{-5}$
$2^{-18}$	1.8813 × 10${}^{-3}$	9.7735 × 10${}^{-4}$	4.1592 × 10${}^{-4}$	1.7348 × 10${}^{-4}$	6.0594 × 10${}^{-5}$
$2^{-21}$	1.8812 × 10${}^{-3}$	9.7734 × 10${}^{-4}$	4.1592 × 10${}^{-4}$	1.7348 × 10${}^{-4}$	6.0594 × 10${}^{-5}$
$2^{-24}$	1.8812 × 10${}^{-3}$	9.7734 × 10${}^{-4}$	4.1592 × 10${}^{-4}$	1.7348 × 10${}^{-4}$	6.0594 × 10${}^{-5}$
$2^{-27}$	1.8812 × 10${}^{-3}$	9.7734 × 10${}^{-4}$	4.1592 × 10${}^{-4}$	1.7348 × 10${}^{-4}$	6.0594 × 10${}^{-5}$
$D^{M,N}$	1.8819 × 10${}^{-3}$	9.7750 × 10${}^{-4}$	4.1598 × 10${}^{-4}$	1.7348 × 10${}^{-4}$	6.0595 × 10${}^{-5}$
$p^{M,N}$	0.94505	1.2326	1.2617	1.5175
$C_p^{M,N}$	1.4909 × 10${}^{-1}$	1.2626 × 10${}^{-1}$	8.7606 × 10${}^{-2}$	5.9570 × 10${}^{-2}$	3.3924 × 10${}^{-2}$

Table 2. Values of $D^{M,N},p^{M,N},$ $C_p^{M,N}$ for $N=128$ and $\alpha =0.9$ for Example 1.

$\mathit{\epsilon }$	Number of Mesh Points M
	40	80	160	320	640
$2^{-4}$	4.8971 × 10${}^{-3}$	2.4621 × 10${}^{-3}$	1.2344 × 10${}^{-3}$	6.1805 × 10${}^{-4}$	3.0923 × 10${}^{-4}$
$2^{-8}$	5.7164 × 10${}^{-3}$	2.8722 × 10${}^{-3}$	1.4396 × 10${}^{-3}$	7.2066 × 10${}^{-4}$	3.6055 × 10${}^{-4}$
$2^{-12}$	6.0399 × 10${}^{-3}$	3.0337 × 10${}^{-3}$	1.5203 × 10${}^{-3}$	7.6098 × 10${}^{-4}$	3.8070 × 10${}^{-4}$
$2^{-16}$	6.1371 × 10${}^{-3}$	3.0826 × 10${}^{-3}$	1.5448 × 10${}^{-3}$	7.7328 × 10${}^{-4}$	3.8686 × 10${}^{-4}$
$2^{-20}$	6.1618 × 10${}^{-3}$	3.0951 × 10${}^{-3}$	1.5511 × 10${}^{-3}$	7.7641 × 10${}^{-4}$	3.8842 × 10${}^{-4}$
$2^{-24}$	6.1692 × 10${}^{-3}$	3.0982 × 10${}^{-3}$	1.5526 × 10${}^{-3}$	7.7719 × 10${}^{-4}$	3.8881 × 10${}^{-4}$
$2^{-28}$	6.1711 × 10${}^{-3}$	3.0990 × 10${}^{-3}$	1.5530 × 10${}^{-3}$	7.7739 × 10${}^{-4}$	3.8891 × 10${}^{-4}$
$2^{-32}$	6.1716 × 10${}^{-3}$	3.0992 × 10${}^{-3}$	1.5531 × 10${}^{-3}$	7.7744 × 10${}^{-4}$	3.8894 × 10${}^{-4}$
$2^{-36}$	6.1717 × 10${}^{-3}$	3.0993 × 10${}^{-3}$	1.5532 × 10${}^{-3}$	7.7745 × 10${}^{-4}$	3.8894 × 10${}^{-4}$
$2^{-40}$	6.1717 × 10${}^{-3}$	3.0993 × 10${}^{-3}$	1.5532 × 10${}^{-3}$	7.7745 × 10${}^{-4}$	3.8895 × 10${}^{-4}$
$2^{-44}$	6.1718 × 10${}^{-3}$	3.0993 × 10${}^{-3}$	1.5532 × 10${}^{-3}$	7.7746 × 10${}^{-4}$	3.8895 × 10${}^{-4}$
$2^{-48}$	6.1718 × 10${}^{-3}$	3.0993 × 10${}^{-3}$	1.5532 × 10${}^{-3}$	7.7746 × 10${}^{-4}$	3.8895 × 10${}^{-4}$
$D^{M,N}$	6.1718 × 10${}^{-3}$	3.0993 × 10${}^{-3}$	1.5532 × 10${}^{-3}$	7.7746 × 10${}^{-4}$	3.8895 × 10${}^{-4}$
$p^{M,N}$	0.99375	0.99673	0.99837	0.99919
$C_p^{M,N}$	4.7690 × 10${}^{-1}$	4.7513 × 10${}^{-1}$	4.7239 × 10${}^{-1}$	4.6913 × 10${}^{-1}$	4.6563 × 10${}^{-1}$

Figure 1. Numerical solution of $y_1$ with $M=10$, $N=128$ and $\epsilon =2^{-15}$ for Example 1.

Figure 2. Numerical solution of $y_2$ with $M=10$ and $N=128$ for Example 1.

Figure 3. Numerical solutions of both components $y_1$ and $y_2$ with $M=10$ and $N=128$ for Example 1.

Figure 4. $Log-logplot$ of the maximum pointwise errors corresponding to Table 1 for Example 1.

Figure 5. $Log-logplot$ of the maximum pointwise errors corresponding to Table 2 for Example 1.

7. Conclusions

To solve numerically a class of m$(m<n)$ singularly perturbed and $n-m$ non-perturbed parabolic equations of the reaction–diffusion boundary value problem, an appropriate combination of standard finite difference schemes using a uniform mesh in the temporal variable and a piecewise uniform mesh in the spatial variable is considered. It is proved that the numerical approximation is uniformly convergent in the maximum norm to the exact solution with respect to the perturbation parameter $\epsilon$. The method reaches an almost second-order convergence in space and almost first-order convergence in time up to a logarithmic factor. A parameter-uniform error bound is obtained for the present method. The numerical results confirm the theoretical outcome of $O\left(M^{-1}+{\left(N^{-1}\ln N\right)}^2\right)$. Additionally, the log-log plots clearly indicate that the error bound has the predicted order.