Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< \epsilon \le \mu \le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $\epsilon$ and $$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < \epsilon \leq \mu \leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order

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

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.Web of Scienc

A numerical layer resolving method for the flow over a curved surface

We investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the stream is parallel to its axis. When the Reynolds number is large, this problem is known to exhibit boundary layers which grow downstream and eventually shows a 3-D character because of the large curvature of the body in the transverse direction. This problem does not have solutions in closed form. We employ a direct layer resolving numerical method that provides solutions that are accurate and robust. To show the direct numerical method is a robust layer-resolving method for the sphere, we provide extensive experimental evidence. We show that all results obtained, such as velocity components and their scaled derivatives, are robust with respect to the viscosity

Grid approximation of a singularly perturbed boundary value problem modelling heat transfer in the case of flow over a flat plate with suction of the boundary layer

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