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

Two optimized high-order compact finite difference (FD) staggered schemes are presented in this communication. Following Holberg's optimization strategy, the least squares problem to minimizing the group velocity (MGV) error, for the fourth- and sixth-order pentadiagonal schemes, is formulated. For a fixed level of group velocity accuracy, the optimized spectrum of wave number and the optimized 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 optimization procedure. Fewer group and phase velocity errors, greater resolution in terms of absolute error, and resolving efficiency, have been achieved by the optimized schemes proposed. High-order accuracy in time is obtained by marching the solution with an optimized 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.