Numerical solution of 2D singularly perturbed reaction–diffusion system with multiple scales
Introduction
Singular perturbation problems (SPP) emerge regularly in a few branches of engineering and applied mathematics. The interesting examples of these system are the reaction–diffusion enzyme model, neutron transport problem and model for saturated flow in fractured porous media. For instance, consider the following model for saturated flow in fractured porous media from [1]: where are the pressure of liquid in the pores of the first and second-order respectively, is the coefficient of compressibility of the liquid and is the viscosity of the liquid. Here and are positive constants, whereas are the porosity of the system of pores of first and second-order respectively, and are the values of the first and second-order porosity at standard pressure. More details on application of these system can be seen in [2, Chapter 12].
In this article, the system of 2D reaction–diffusion initial–boundary-value problem (IBVP) on the domain is considered: where the spatial differential operator is given by The diffusion and reaction coefficients are given by and , respectively. Here, we assume that is strongly diagonally dominant with In particular, assume that there exist and such that
In the past, SPPs with boundary and interior layers have been studied and analyzed with a wide range of numerical schemes. For instances; Natesan et al. [3], [4] developed numerical approximation for turning point problem and weak boundary layer problems. A parallel boundary-value technique has been developed for SPP by Vigo-Aguiar and Natesan in [5]. Simos and Vigo-Aguiar studied the modified Runge Kutta method for the numerical solution of the Schrödinger equation in [6]. Vigo-Aguiar and Natesan [7] analyzed an efficient numerical method for SPP by splitting the model problem in the system of two first-order differential equations. Ramos et al. [8] solved nonlinear singularly perturbed problems by using a non-standard algorithm.
A brief literature about the computational methods for the singularly perturbed system of reaction–diffusion problems (SPSRDP) is presented here. The fitted difference scheme for SPSRDP involving small diffusion parameters of different values is considered in [9], [10]. To improve the accuracy of the numerical approximation of the previous article, a special approach for the error estimate of the numerical method is followed in [11], [12]. In [13], optimal error estimate for SPSRDP by using mesh equidistribution technique is derived. For time-dependent one-dimensional SPSRDP, a uniform convergent numerical scheme is proposed in [14]. A numerical scheme for time-dependent 1D SPSRDP is studied in [15]. Recently, the authors [16] studied upwind based finite difference scheme for time-dependent singularly perturbed system of convection–diffusion problem (SPSCDP) which produces first-order accurate results. To improve the order of convergence of the approximate solution, we applied extrapolation idea in [17].
There are very few articles available in the literature, which deals with the numerical solution of singularly perturbed 2D parabolic IBVPs. In [18], a monotone finite difference scheme is considered to approximate the solution of time-dependent SPP of convection–diffusion type with implicit-Euler method as time integrators, whereas a higher-order numerical method is proposed in [19]. An ADI numerical scheme for time-dependent SPP of degenerate type is studied in [20]. Again, for system of PDE in two-dimension, very few articles are available in the literature: In [21], [22], the classical central difference scheme is considered to solve singularly perturbed system of elliptic BVPs. Recently, the authors proposed numerical methods for time dependent SPSCDP involving single diffusion parameters in [23]. Numerical simulations for time dependent SPSCDP and SPSRDP involving single parameters have been analyzed in [24], [25]. Therefore, it is always interesting to study singularly perturbed systems in more generalized sense, i.e., model problems with diffusion parameters of different magnitudes.
The primary intent of this study is to develop a robust finite difference method and analyze the error estimates for 2D parabolic SPSRDP involving distinct diffusion parameters. Prior knowledge about the behavior of the exact solution is required to obtain the error estimates. Because of the multi-scale nature of the model problem, it is difficult to extract the analytical properties of the exact solution. For this reason, here, we use the Shishkin-type decomposition of the exact solution and obtain the bounds for the derivatives of the solution for regular and layer components. Initially, we discretize the spatial derivatives by employing the central difference scheme on a layer-adapted Shishkin mesh. Later on, we approximate the resulting stiff initial-value problems by the implicit-Euler scheme. We derive the error estimate, which is almost second-order accurate in the spatial variables and first-order in the temporal variable.
This article is presented as follows: In Section 2, we discuss some results concerning the analytic behavior of the exact solution and its derivatives and later on a Shishkin-type decomposition has been established. In Section 3, the spatial semidiscretization has been introduced and we provide its truncation error analysis. For the time discretization, the implicit-Euler scheme has been used and the convergence result of the fully discrete scheme has been established in Section 4. In Section 5, we carried out some numerical experiments to show the accuracy and efficiency of the proposed method, and validate the theoretical error estimates. Finally, we concluded our work in Section 6.
Throughout this article, denotes a generic positive constant, which is independent of and . We denote the partial derivative of with respect to by , whereas denotes the time derivative.
Next, we introduce some notations for barrier functions. Let us define the following notations where is defined in (1.3) also assume that , from a practical point of view, this is always true.
For a vector-valued function , set and define the maximum norm as . And define the maximum norm on , for a mesh function by For the mesh function , define the maximum norm as .
Section snippets
The continuous problem
In this section, we discuss the behavior of the exact solution, the bounds on the derivatives of the solution, the solution decomposition and their respective derivative bounds.
We assume that the data of the 2D system of parabolic IBVPs (1.1) are sufficiently smooth functions and also satisfy sufficient compatibility conditions in order to have a unique solution . For instance, consider the typical smoothness assumptions for the source term and the initial value
The spatial semidiscretization
In this section, we discretize the spatial derivatives of the model problem (1.1) by using the central difference scheme on the layer-adapted piecewise-uniform Shishkin mesh. At the end, we will discuss the error analysis for the spatial discretization.
We discretize the model problem (1.1) on a rectangular mesh (here, we take same number of mesh points for both the spatial variables and ). We define the mesh transition parameters by
The time discretization
We use the implicit-Euler method to discretize the semidiscrete problem (3.1). Later, the convergence analysis of the fully discrete scheme has been established.
Numerical experiments
To illustrate the accuracy of the numerical method and the theoretical results of the error analysis, we present some numerical results.
Here, we have discussed the implementation of the numerical approximation in Matlab. We express the fully discrete scheme (4.2) by the following system of linear algebraic equations: where is given in (3.3)–(3.4). In the matrix–vector form, we can write the system
Concluding remarks
In this research paper, we have considered a finite difference scheme to obtain the numerical solution of singularly perturbed system of 2D parabolic reaction–diffusion IBVPs on the piecewise-uniform Shishkin mesh. For discretizing the spatial derivatives, second-order central difference scheme has been used and then the implicit-Euler scheme has been used for the time derivative. The proposed numerical scheme is of first-order convergence in the temporal direction and almost second-order
CRediT authorship contribution statement
Maneesh Kumar Singh: Software, Validation, Writing - original draft. Srinivasan Natesan: Conceptualization, Methodology, Writing - review & editing.
Acknowledgments
The authors express their sincere thanks to the anonymous referees for carefully reading the manuscript and providing suitable suggestions to improve the presentation. The first author gratefully acknowledges financial support from the National Board for Higher Mathematics (NBHM), Government of India .
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