On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations (Communications in Computational Physics, 2018)
Implicit time integration schemes are popular because their relaxed stabilityconstraints can result in better computational efficiency. For time-accurate unsteadysimulations, it has been well recognized that the inherent dispersion and dissipationerrors of implicit Runge-Kutta schemes will reduce the computational accuracy forlarge time steps. Yet for steady state simulations using the time-dependent governingequations, these errors are often overlooked because the intermediate solutions are ofless interest. Based on the model equation dy/dt = (µ+iλ)y of scalar convection diffusionsystems, this study examines the stability limits, dispersion and dissipation errorsof four diagonally implicit Runge-Kutta-type schemes on the complex (µ+iλ)∆tplane. Through numerical experiments, it is shown that, as the time steps increase,the A-stable implicit schemes may not always have reduced CPU time and the computationsmay not always remain stable, due to the inherent dispersion and dissipationerrors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate theconvergence rate, and the dispersion errors may cause large oscillations of the numericalsolutions. These errors, especially those of high wavenumber components, growat large time steps. They lead to difficulty in the convergence of the numerical computations,and result in increasing CPU time or even unstable computations as the timestep increases. It is concluded that an optimal implicit time integration scheme forsteady state simulations should have high dissipation and low dispersion.