A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

Abstract

This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings. IntroductionWhen solving numerically time-dependent singularly perturbed problems, it is customary to consider dimension splitting: The problems are semidiscretized in time (for example by using the classical backward Euler or Crank-Nicolson scheme). Then, at each time level, a set of stationary problems is solved using a suitably designed numerical method. Singularly perturbed problems involve a (perturbation) parameter which multiplies the highest derivative term in the model-equation of the problem. The solution to such problems is characterized by layer regions which are narrow parts of the domain over which the solution undergoes abrupt changes. It is well known that classical methods are not appropriate when the perturbation parameter becomes small unless very fine meshes are used for spatial discretization. However, this approach has two side effects: it increases the round-off error and the computational cost. There is a vast literature about non-classical numerical methods. In the context of finite differences, we can group these methods into two classes: the class of fitted mesh methods and the class of fitted operator methods. Both these types of methods have been used to solve stationary singularly perturbed problems in one and several dimensions. As examples, see Linß and Stynes (1999), Lubuma andPatidar (2006), Miller et al. (1996), Munyakazi and Patidar (2010a,b, 2012), Patidar (2005, 2007, Roos et al. (1996), Shishkin (1986. It should be noted that the discovery/development of fitted mesh methods is anterior to that of the fitted operator ones. The analysis of the latter is simpler due to the fact that they are based on uniform meshes unlike the former where non-uniform meshes are designed.

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

“…Applications of these NSFDMs for singularly perturbed problems can be seen in the work of Kadalbajoo et al (2006), Lubuma and Patidar (2006;2007c;2007a;2007b), Munyakazi and Patidar (2010), Patidar and Sharma (2006a;2006b), or Patidar (2008. However, an exhaustive account of work that use such methods is provided in the survey article by Patidar (2005).…”

Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data

“…The nature of the interior layer depends on the sign of the convection coefficient ( ) a x on either side of the line of discontinuity [13][14][15][16][17][18]. Thus, to stress the existence of strong interior layers, consider the following particular condition…”

Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

“…The numerical solution of these problems with smooth data and nonsmooth data are also considered in several journals and proceedings. Singularly perturbed parabolic problems are also considered for numerical analysis in Bansal et al and Munyakazi and Patidar . An almost first‐order accurate solution for two‐parameter singulary perturbed time‐dependent problems with smooth data can be seen in O'Riordan et al on a piecewise uniform mesh.…”

“…Singularly perturbed parabolic problems are also considered for numerical analysis in Bansal et al and Munyakazi and Patidar. 20,21 An almost first-order accurate solution for two-parameter singulary perturbed time-dependent problems with smooth data can be seen in O'Riordan et al 9 on a piecewise uniform mesh. In Kadalbajoo and Yadaw, 22 the authors proposed a combination of finite element method in space with implicit Euler method in temporal direction using Rothe's method and obtained an almost second-order accuracy in space.…”