Asymptotic Solutions of 1-D Singularly Perturbed Convection-Diffusion Equations with a Turning Point: The Compatible Case

Applied MathematicsIn this article, we consider a convection-diffusion equation with a small diffusion coefficient . It is a version of a linearized Navier-Stokes equation. Due to the small parameter multiplied to the highest order of differential operators, the so-called turning point transition layers are displayed where flows in opposite directions collide. For example, a turning point can be observed where the Kuroshio and Kurile Currents meet, from opposite directions, in the North Pacific. Unlike boundary layers, turning point transition layers occur where the convective flows collide and more delicate analysis is necessitated. Especially, we consider a single turning point with multiple-orders in one-dimensional spaces and provide sharp estimations for the solution with compatible conditions. A main difficulty when solving the problem arises from the fact that the diffusion coefficient is very small in comparison with other terms and it causes a singularity in the solution. We use the asymptotic analysis that is different from typical methods in the singular perturbation problem considered here. The matching technique has been typically used, but this method brings about the difficulty in constructing a globally matched solution. Our method is relatively easy to analyze, and turning point transition layers are systematically and easily constructed.ope

A uniformly convergent quadratic B-spline based scheme for singularly perturbed degenerate parabolic problems.

[EN]In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme

A fully discrete ε-uniform method for convection-diffusion problem on equidistant meshes

For a singularly-perturbed two-point boundary value problem, we propose an ε-uniform finite difference method on an equidistant mesh which requires no exact solution of a differential equation. We start with a full-fitted operator method reflecting the singular perturbation nature of the problem through a local boundary value problem. However, to solve the local boundary value problem, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. We further study the convergence properties of the numerical method proposed and prove it nodally converges to the true solution for any ε

An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers

In this paper we consider a 1D parabolic singularly perturbed reaction-convection-diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter e2) and the convection term (multiplied by the parameter µ) in the differential equation (e¿(0, 1], µ¿0, 1], µ=e). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in e and µ, having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed