Alan F. Hegarty

Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter @e, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates @e-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the @e-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.

Numerical solution of a convection diffusion problem with Robin boundary conditions

We consider a one-dimensional steady-state convection dominated convection-diffusion problem with Robin boundary conditions. We show, both theoretically and with numerical experiments, that numerical solutions obtained using an upwind finite difference scheme on Shishkin meshes are uniformly convergent with respect to the diffusion coefficient.

A note on iterative methods for solving singularly perturbed problems using non-monotone methods on Shishkin meshes

Abstract Non-monotone methods with Shishkin meshes are employed in obtaining finite difference schemes for solving a linear two-dimensional steady state convection–diffusion problem. Preconditioners are used that significantly reduce the number of iterations of the linear solver. Computational results for a Galerkin method are presented which indicate parameter robust, super-linear orders of convergence.

A note on fitted operator methods for a laminar jet problem

We consider the classical problem of a two-dimensional laminar jet of incompressible fluid flowing into a stationary medium of the same fluid [H. Schlichting, Boundary-Layer Theory, McGraw-Hill, 1979]. The equations of motion are the same as the boundary layer equations for flow over an infinite flat plate, but with different boundary conditions. It has been shown [A.R. Ansari et al., Parameter robust numerical solutions for the laminar freejet, submitted] that using an appropriate piecewise uniform mesh, numerical solutions together with their scaled discrete derivatives are obtained which are parameter (i.e., viscosity v) robust with respect to both the number of mesh nodes and the number of iterations required for convergence. We prove that there do not exist fitted operator schemes which converge v-uniformly if the fitting coefficients are independent of the problem data.

An experimental technique for computing parameter-uniform error estimates for numerical solutions of singular perturbation problems, with an application to Prandtl's problem at high Reynolds number

In this paper we describe an experimental technique for computing realistic values of the parameter-uniform order of convergence and error constant in the maximum norm associated with a parameter-uniform numerical method for solving singularly perturbed problems. We employ the technique to compute Reynolds-uniform error bounds in the maximum norm for the numerical solutions generated by a fitted-mesh upwind finite difference method applied to Prandtls problem arising from laminar flow past a thin flat plate. Thus we illustrate the efficiency of the technique for finding realistic parameter-uniform error bounds in the maximum norm for the approximate solutions generated by numerical methods for which no theoretical error analysis is available.

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.

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.

Numerical results for advection‐dominated heat transfer in a moving fluid with a non‐slip boundary condition

This paper is concerned with the laminar transfer of heat by forced convection where the velocity profile is taken to be parabolic. In the advection dominated case the problem is described mathematically by a singularly perturbed boundary value problem with a non‐slip condition. It has been established both theoretically and computationally that numerical methods composed of upwind finite difference operators on special piecewise uniform meshes have the property that they behave uniformly well, regardless of the magnitude of the ratio of the advection term to the diffusion term. A variety of choices of special piecewise uniform mesh is examined and it is shown computationally that these lead to numerical methods also sharing this property. These results validate a previous theoretical result which is quoted.

Numerical Results for Singularly Perturbed Convection-Diffusion Problems on an Annulus

Numerical methods for singularly perturbed convection-diffusion problems posed on annular domains are constructed and their performance is examined for a range of small values of the singular perturbation parameter. A standard polar coordinate transformation leads to a transformed elliptic operator containing no mixed second order derivative and the transformed problem is then posed on a rectangular domain. In the radial direction, a piecewise-uniform Shishkin mesh is used. This mesh captures any boundary layer appearing near the outflow boundary. The performance of such a method is examined in the presence or absence of compatibility constraints at characteristic points, which are associated with the reduced problem.