Superconsistent collocation methods with applications to convection-dominated convection–diffusion equations

Abstract

Superconsistency, as it applies to collocation schemes for linear partial differential equations, is given what we believe to be a new and precise definition. We introduce new superconsistent collocation schemes for linear partial differential equations and illustrate their use for the solution of two and three dimensional steady convection-dominated convection–diffusion equations, where the exact solutions exhibits exponential boundary layers near the outflow boundary of the convection velocity field. The numerical results reveal that our methods are more accurate than both a centred finite difference scheme and an existing upwinded scheme due to Fatone et al. (2005) over a range of convection velocities and grid sizes.