New optimized fourth-order compact finite difference schemes for wave propagation phenomena

Two optimized fourth-order compact centered finite difference schemes are presented in this paper. By minimizing, over a range of the wave numbers domain, the variations of the phase speed with the wave number, an optimization least-squares problem is formulated. Hence, solving a linear algebraic system, obtained by incorporating the relations between the coefficients for the fourth-order three-parameter family schemes, the corresponding well-resolved wave number domains, and the related optimized coefficients, for two levels of accuracy, are analytically evaluated. Several dispersion comparisons, including the asymptotic behavior between the proposed and other existing optimized pentadiagonal fourth-order schemes, are presented and discussed. The schemes applicable directly on the interior nodes, are associated with a set of fourth-order boundary closure expressions. By adopting a fourth-order six-stage optimized Runge–Kutta algorithm for time marching, the stability bounds, the global errors, and the computational efficiency, for the fully discrete schemes, are examined. The performances of the presented schemes are tested on benchmark problems that involve both the one-dimensional linear convection and the one-dimensional nonlinear shallow water equations. Finally, the one-dimensional schemes are extended to two dimensions and, using the two dimensional shallow water equations, classical applications are presented. The results allow us to propose, as the ideal candidate for simulating wave propagation problems, the scheme which corresponds to the strict level of accuracy with the maximum resolution over a narrow wave number space.