Eugene O’Riordan

Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems

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 is 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.

A coupled system of singularly perturbed parabolic reaction-diffusion equations

In this paper systems with an arbitrary number of singularly perturbed parabolic reaction-diffusion equations are examined. A numerical method is constructed for these systems which involves an appropriate layer-adapted piecewise-uniform mesh. The numerical approximations generated from this method are shown to be uniformly convergent with respect to the singular perturbation parameters. Numerical experiments supporting the theoretical results are given.

Opposing flows in a one dimensional convection-diffusion problem

In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator to construct a parameter-uniform numerical method for this class of singularly perturbed problems.

On Numerical Experiments with Central Difference Operators on Special Piecewise Uniform Meshes for Problems with Boundary Layers

Singularly perturbed second order elliptic equations with boundary layers are considered. Numerical methods composed of central difference operators on special piece-wise uniform meshes are constructed for the above problems. Numerical results are obtained which show that these methods give approximate solutions with error estimates that are independent of the singular perturbation parameter.

Patches of Finite Elements for Singularly-Perturbed Diffusion Reaction Equations with Discontinuous Coefficients

We present a numerical method for solving Diffusion Reaction equations on two completely overlapping unstructured meshes which reduces the requirements on mesh generation software when strong local refinement is needed to capture features of the solution that appear on different scales.

On the Design of Piecewise Uniform Meshes for Solving Advection-Dominated Transport Equations to a Prescribed Accuracy

The numerical performance of numerical methods specifically designed for singularly perturbed partial differential equations is examined. Numerical methods whose solutions have an accuracy independent of the small parameter are called e-uniform methods. In this paper, the advantages of using an e-uniform numerical method are discussed.

Parameter-uniform numerical method for singularly perturbed convection-diffusion problem on a circular domain

A linear singularly perturbed elliptic problem, of convection-diffusion type, posed on a circular domain is examined. Regularity constraints are imposed on the data in the vicinity of the two characteristic points. The solution is decomposed into a regular and a singular component. A priori parameter-explicit pointwise bounds on the partial derivatives of these components are established. By transforming to polar co-ordinates, a monotone finite difference method is constructed on a piecewise-uniform layer-adapted mesh of Shishkin type. Numerical analysis is presented for this monotone numerical method. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established.

Convergence Outside the Initial Layer for a Numerical Method for the Time-Fractional Heat Equation

In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order \(\delta \) where \(0<\delta <1\). It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time \(t=0\); this reduces the global order of convergence of the finite difference method to \(O(h^2+\tau ^\delta )\), where h and \(\tau \) are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from \(t=0\). This motivates us to investigate if the finite difference method is more accurate away from \(t=0\). Numerical experiments with various non-smooth and incompatible initial conditions show that, away from \(t=0\), one obtains \(O(h^2+\tau )\) convergence.

Interior Layers in Singularly Perturbed Problems

To construct layer adapted meshes for a class of singularly perturbed problems, whose solutions contain boundary layers, it is necessary to identify both the location and the width of any boundary layers present in the solution. Additional interior layers can appear when the data for the problem is not sufficiently smooth.In the context of singularly perturbed partial differential equations, the presence of any interior layer typically requires the introduction of a transformation of the problem, which facilitates the necessary alignment of the mesh to the trajectory of the interior layer. Here we review a selection of published results on such problems to illustrate the variety of ways that interior layers can appear.

Numerical approximation of solution derivatives of singularly perturbed parabolic problems of convection-diffusion type

Numerical approximations to the solution of a linear singularly perturbed parabolic problem are generated using a backward Euler method in time and an upwinded finite difference operator in space on a piecewise-uniform Shishkin mesh for a convectiondiffusion problem. A proof is given to show first order convergence of these numerical approximations in appropriately weighted C-norm. Numerical results are given to support the theoretical error bounds. The analysis is also applied to singularly perturbed problems of reaction-diffusion type. AMS Classification: 65M15,65M12, 65M06