Numerical experiments for advection–diffusion problems in a channel with a 180° bend

Abstract

In this paper we solve numerically two singularly perturbed linear convection–diffusion problems for heat transfer in a fluid with an assumed flow field in the neighbourhood of a 180° bend in a channel. In the first problem the theoretical solution has a parabolic boundary layer and in the second problem there is both a parabolic and a regular boundary layer in the solution. The numerical method uses piecewise uniform fitted meshes condensing in a neighbourhood of each boundary layer and a standard upwind finite difference operator satisfying a discrete maximum principle. The numerical results confirm computationally that the method is ε-uniform in the sense that the rate of convergence and the error constant of the method are independent of the singular perturbation parameter ε, where ε denotes the reciprocal of the Péclet number of the fluid. This ε-uniform behaviour is obtained only when an appropriate piecewise uniform fitted mesh is constructed for each boundary layer. This is confirmed by several additional computations on meshes which do not fulfill this requirement.