An improved dealiasing scheme for the fourth-order Runge-Kutta method: Formulation, accuracy and efficiency analysis

Abstract

In this paper, the Random Phase Shift Method (RPSM) dealiasing scheme has been developed with the classical fourth-order explicit Runge-Kutta (RK4) method. This scheme is implemented in different benchmark problems to verify its numerical accuracy and computational efficiency where strong gradients are present in the solution. The propagation of aliasing errors through the substeps of RK4 is derived to show the existence of the residual aliasing error terms which results in mild smoothing effect without dissipating the small-scale flow structures. Smoothness and numerical stability in the solutions obtained from the RPSM scheme also remain well preserved even at under-resolved conditions. Numerical results agree well with the analytical and the computed solutions from previous studies. RPSM scheme shows a slight delay in the formation of numerical singularity for the inviscid flows but the filtering-based schemes suffer from early blow-up problem. We observe that this scheme displays better resolving ability than higher-order exponential smoothing spectral filter scheme in capturing the strong fronts accurately even at just resolved spatial grid resolutions. Three-dimensional truncation-based dealiasing scheme, spherical truncation (SPT) shows vortices generated due to the parasitic currents in the solution of the inviscid three-dimensional Taylor Green (TG) vortex flows. RPSM displays only the accurate isocontours of vortical field at nearly same computational expenses as the SPT scheme. � 2020 John Wiley & Sons, Ltd.