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

Abstract

The effect of using grid adaptation on the numerical solution of model convection-diffusion equations with a conservation form is studied. The grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc-length of the solution. In particular, a parameter-dependent monitor function is introduced which incorporates fixed meshes, approximate arc-length equidistribution, 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's equation in the limit of small viscosity, is studied. Singular perturbation and bifurcation techniques are used to analyze 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 fixed mesh and that, furthermore, the spurious solutions can be stable for an appropriate moving mesh method.