A strongly S-stable low-dissipation and low-dispersion Runge-Kutta scheme for convection diffusion systems

Abstract

Since the A-stability and the order of accuracy of time integration methods have been proven insufficient to guarantee the time-accurate simulations of convection diffusion systems, optimized implicit Runge-Kutta schemes with enhanced stability have been developed for stiff problems, and separately those with low-dispersion low-dissipation errors have been proposed for sensitive wave propagation phenomena. However, an implicit Runge-Kutta method is ideally preferred to address both disparate stiffness and various wave propagation characteristics that may be unknown in advance but often co-exist in complex systems, such as turbulent flows with multi-physical phenomena. Therefore, an optimized three-stage second-order diagonally implicit Runge-Kutta scheme with the strong S-stability and low-dispersion low-dissipation errors is derived in this study. Numerical benchmark tests show that overall this newly derived scheme has comprehensively the best performance among the well-known second-order implicit Runge-Kutta schemes. It reaches a second-order accuracy and produces accurate solutions for wave propagation phenomena, but is also stiffly first-order accurate and remains stable with very large time steps for strongly stiff problems.