Generalized High Order Compact Schemes

Dr. Bill Spotz
Program Manager
Office of Science, Advanced Scientific Computing Research
US Dept of Energy

Abstract: "High order compact" (HOC) finite difference methods refer to methods that obtain an order of accuracy that is higher than typically obtained on a given stencil of grid points. As one example, the standard central difference expressions for the first and second derivatives on three-point stencils are second order, but can be improved to fourth order on the same stencil by changing from an explicit to an implicit formulation. The compact, high order schemes in this talk obtain their higher accuracy by including information from the governing partial differential equation (either the equation itself or its derivatives) in the formulation. This idea dates back to Lax-Wendroff, but the approach has been expanded from time derivatives to spatial derivatives in 1D, 2D and 3D, for the Poisson equation and convection-diffusion, as well as other PDEs of interest. The advantages include not only higher order accuracy, but also the serendipitous suppression of artificial oscillations. The disadvantages include great algebraic complexity that is prone to errors and limited flexibility in the grids that can be used, thus limiting the class of problems that can be solved. This talk will focus on the Generalized High Order Compact (GHOC) scheme, which attempts to address these disadvantages of the HOC scheme. It is an extension of the Generalized Finite Difference Method, which works on unstructured grids, and simplifies the algebra by transferring tedious elements of the method derivation to the computer. I will cover the derivation of GHOC, as well as error analysis and the results of numerical experiments.