Uniformly Accurate Finite Difference Schemes forp-Refinement

Abstract

The accuracy deficiency of high-order conventional finite difference schemes in the high-frequency range is attributed to the approximation property of a power polynomial interpolation over equispaced nodes. To improve the numerical representation of high-frequency solutions, a new family of high-order finite difference schemes is constructed by trigonometric polynomial interpolation. As expected, these new difference schemes can resolve a wider frequency range than conventional finite difference schemes with the same stencil. In particular, they remove the possibility of Runge's phenomenon for uniform mesh. The recently proposed optimized difference schemes are indeed types of mixed difference schemes which make a compromise between power and trigonometric polynomial interpolation.