On a singularly perturbed, coupled parabolic-parabolic problem

Applicable Analysis, 2006

A singularly perturbed convection-diffusion equation with constant coefficients is considered in a half plane, with Dirichlet boundary conditions. The boundary function has a specified degree of regularity except for a jump discontinuity, or jump discontinuity in a derivative of specified order, at a point. Precise pointwise bounds for the derivatives of the solution are obtained. The bounds show both the strength of the interior layer emanating from the point of discontinuity and the blowup of the derivatives resulting from the discontinuity, and make precise the dependence of the derivatives on the singular perturbation parameter.

Advances in Computational Mathematics, 2009

A system of two coupled singularly perturbed convection-diffusion ordinary differential equations is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh are shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm.

Cornell University - arXiv, 2022

In this article, we have considered a time-dependent two-parameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters and µ multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss the problem for ratio µ 2. We used an upwind finite difference approach on a Shishkin-Bakhvalov mesh in the space and the Crank-Nicolson method in time on uniform mesh. At the point of discontinuity, a three-point formula was used. This method is uniformly convergent with second order in time and first order in space. Shishkin-Bakhvalov mesh provides first-order convergence; unlike the Shishkin mesh, where a logarithmic factor deteriorates the order of convergence. Some test examples are given to validate the results presented.

Cornell University - arXiv, 2022

We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.

Acta Applicandae Mathematicae, 2000

We consider a singularly perturbed convection-diffusion equation, − u + v · ∇u = 0, defined on a half-infinite strip, (x, y) ∈ (0, ∞) × (0, 1) with a discontinuous Dirichlet boundary condition: u(x, 0) = 1, u(x, 1) = u(0, y) = 0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter → 0 + (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r → 0 + (with fixed ). It is shown that the first term of the expansion at = 0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x, y) is approximated by a linear function of the polar angle at the point of discontinuity (0, 0). : 35C20, 41A60.

Mathematical Modelling and Analysis, 2015

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the para...

Journal of Scientific Computing, 2022

A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A domain decomposition method is used, which uses a rectangular grid outside the boundary layer and a Shishkin mesh, aligned to the curvature of the outflow boundary, near the boundary layer. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.

Mathematics in Industry, 2010

We show the importance of the error function in the approximation of the solution of singularly perturbed convection-diffusion problems with discontinuous boundary conditions. It is observed that the error function (or a combination of them) provides an excellent approximation and reproduces accurately the effect of the discontinuities on the behaviour of the solution at the boundary and interior layers.

ceunet.ceu.hu

Abstract. In this paper we investigate a model for convection-diffusion-reaction processes in which there is a small parameter ε > 0 representing the diffusion coefficient in a sub-domain of the spatial domain. Specifically, the problem is (S∈), (IC∈), (BC∈), (TC∈) formulated below. ...

Studies in Applied Mathematics, 2004

We consider a singularly perturbed convection-diffusion equation, − u + v · ∇u = 0, defined on two domains: a quarter plane, (x, y) ∈ (0, ∞) × (0, ∞), and an infinite strip, (x, y) ∈ (−∞, ∞) × (0, 1). We consider for both problems discontinuous Dirichlet boundary conditions: u(x, 0) = 0 and u(0, y) = 1 for the first one and u(x, 0) = χ [a,b] (x) and u(x, 1) = 0 for the second. For each problem, asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) when the singular parameter → 0 + (with fixed distance r to the discontinuity points of the boundary condition) and (b) when that distance r → 0 + (with fixed ). It is shown that in both problems, the first term of the expansion at = 0 is an error function or a combination of error functions. This term characterizes the effect of the discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the discontinuities of the boundary condition, the solution u(x, y) of both problems is approximated by a linear function of the polar angle at the discontinuity points.