Robert Struijs
A Basis for Discretizations
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ABSTRACT. We discuss approximations of a given order of accuracy to first and higher derivatives on a structured grid in N dimensions using a basis of stencils. The basis of stencils follows from a truncated Taylor series expansion of the nodal values when the extent and the consistency of the approximation are imposed, together with the order of the error.
The approximations include points which do not lie on grid lines passing through the point of discretization. Examples of such discretizations are diagonal discretizations, and a generalization of the formulation of Hildebrand.
The basis of stencils is a convenient tool for generating and optimizing discretizations. The optimization is for the truncation error, or for stability and time step. In problems with a preferential direction such as the advection equation, the approximations are directional discretizations. The reduction of directional components of the truncation error leads to streamline approximations. Significant error reductions are possible, but they require a very regular grid. The stability and the time step in three dimensions increases threefold. The optimizations also apply to unstructured grids, but the computational advantage in three dimensions is marginal. The limiting functions for the grid-based derivatives are coupled, which complicates their solution.