On the Solution of Convection-Diffusion Boundary Value Problems Using Equidistributed Grids

The eeect of using grid adaptation on the numerical solution of model convection-diiusion equations with a conservation form is studied. The grid adaptation technique studied is based on moving a xed number of mesh points to equidistribute a generalisation of the arc-length of the solution. In particular a parameter dependent monitor function is introduced which incorporates xed meshes, approximate arc-length equi-distribution and equidistribution of the absolute value of the solution, in a single framework. Thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values. A class of singularly perturbed problems , including Burgers' equation in the limit of small viscosity, is studied. Singular perturbation and bifurcation techniques are used to analyse the solution of the discretized equations and numerical results are compared with the results from the analysis. Computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory. It is shown that equidistribution does not remove spurious solutions present on a xed mesh and that, furthermore, the spurious solutions can be stable for an appropriate moving mesh method.