A Model of Porous Catalyst Accounting for Incipiently Non-isothermal Effects*

An approximate model accounting for incipiently non-isothermal effects is derived from a well-known model of porous catalyst for appropriate, realistic limiting values of the parameters. In this limit, the original model is a singularly perturbed, m-D reaction–diffusion system, and the approximate model is given by the m-D heat equation with nonlinear boundary condition, coupled with infinitely many (ifm2) 1-D semilinear parabolic equations, one for each point of the boundary of the spatial domain. Some limiting cases are still considered in the approximate model that lead to further simplifications

A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm

Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques

A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Second order parameter-uniform convergence for a finite difference method for a partially singularly perturbed linear parabolic system

A linear system of $n$ second order differential equations of parabolic reaction-diffusion type with initial and boundary conditions is considered. The first $k$ equations are singularly perturbed. Each of the leading terms of the first $m$ equations, $mleq k$, is multiplied by a small positive parameter and these parameters are assumed to be distinct. The leading terms of the next $k-m$ equations are multiplied by the same perturbation parameter $varepsilon_m$. Since the components of the solution exhibit overlapping layers, Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that in the maximum norm the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable, uniformly with respect to all of the parameters