On richardson extrapolation for finite difference methods on regular grids

Summary Difference solutions of partial differential equations can in certain cases be expanded by even powers of a discretization parameterh. If we haven solutions corresponding to different mesh widthsh ₁,...,h _n we can improve the accuracy by Richardson extrapolation and get a solution of order 2n, yet only on the intersection of all grids used, i.e. normally on the coarsest grid. To interpolate this high order solution with the same accuracy in points not belonging to all grids, we need 2n points in an interval of length (2n–1)h ₁.This drawback can be avoided by combining such an interpolation with the extrapolation byh. In this case the approximation depends only on grid points in an interval of length 3/2h ₁. The length of this interval is independent of the desired order.By combining this approach with the method of Kreiss, boundary conditions on curved boundaries can be discretized with a high order even on coarse grids.This paper is based on a lecture held at the 5th Sanmarinian University Session of the International Academy of Sciences San Marino, at San Marino, 1988-08-27-1988-09-05