Holberg's optimisation for high-order compact finite difference staggered schemes

Abstract

Two optimised high-order compact finite difference (FD) staggered schemes are presented in this communication. Following Holberg's optimisation strategy, the least squares problem to minimising the group velocity (MGV) error, for the fourth- and sixth-order pentadiagonal schemes, is formulated. For a fixed level of group velocity accuracy, the optimised spectrum of wave number and the optimised coefficients for the schemes, are analytically evaluated. The spectral accuracy of these schemes has been verified by several comparisons with the FD staggered schemes obtained following Kim and Lee's (1996) optimisation procedure. Fewer group and phase velocity errors, greater resolution in terms of absolute error and resolving efficiency have been achieved by the optimised schemes proposed. High-order accuracy in time is obtained by marching the solution with an optimised Runge–Kutta scheme. Next, the comparison in terms of the number of grid points per wavelength required to achieve a standard accuracy for distances expressed in terms of the number of wavelengths travelled is presented. Numerical results from benchmark tests for the one-dimensional shallow water equations are presented.

Keywords:

• finite difference schemes
• staggered formulations
• optimisation method
• dispersion
• group velocity