A Wavenumber Based Extrapolation and Interpolation Method for Use in Conjunction with High-Order Finite Difference Schemes

The errors incurred in using extrapolation and interpolation in large scale computations are analyzed and quantified in the wavenumber space. If a large extrapolation stencil is used, the errors in the low wavenumbers can be significantly reduced. However, the errors in the high wavenumbers are, at the same time, greatly increased. The opposite is true if the stencil size is reduced. Based on the wavenumber analysis, an optimized extrapolation and interpolation method is proposed. The optimization is carried out over a selected band of wavenumbers. It is known that extrapolation often leads to numerical instability. The instability is the result of large error amplification in the high wavenumber range. To reduce the tendency to trigger numerical instability, it is proposed that an extra constraint be imposed on the optimized extrapolation method. The added constraint aims to reduce error amplification over the high wavenumbers. Numerical examples are provided to illustrate that accurate and stable numerical results can be obtained in large scale simulation using a high-order finite difference scheme and the proposed optimized extrapolation method. When the same problems are recomputed using the familiar high-order polynomials extrapolation method in the Lagrange form, in one case the numerical results are plagued by large errors and ultimately instability. In another problem, it is found that the use of the Lagrange polynomials extrapolation method would lead immediately to numerical instability.