Efficient implementation of high order methods for the advection–diffusion equation Original Research Article

A new approach to designing high order – defined here to exceed third – accurate methods has been developed and tested for a linear advection–diffusion equation in one and two dimensions. The systematic construction of progressively higher order spatial approximations is achieved via a modified equation analysis, which allows one to determine the computational stencil coefficients appropriate for a desired accuracy order. A distinguishing desirable property of the developed method is solution matrix bandwidth containment, i.e. bandwidth always remains equal to that of the second-order discretization. Numerical simulations compare performance of the developed fourth- and sixth-order methods to that of the linear and bilinear basis Galerkin weak statement formulations in one and two dimensions, respectively. Uniform mesh refinement convergence results confirm the order of truncation error for each method. High order approximations are shown to require significantly fewer nodes to accurately resolve solution gradients for convection dominated problems.