@Article{CiCP-20-2,
author = {},
title = {On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations},
journal = {Communications in Computational Physics},
year = {2016},
volume = {20},
number = {2},
pages = {486--511},
abstract = {<p style="text-align: justify;">Implicit time integration schemes are popular because their relaxed stability
constraints can result in better computational efficiency. For time-accurate unsteady
simulations, it has been well recognized that the inherent dispersion and dissipation
errors of implicit Runge-Kutta schemes will reduce the computational accuracy for
large time steps. Yet for steady state simulations using the time-dependent governing
equations, these errors are often overlooked because the intermediate solutions are of
less interest. Based on the model equation dy/dt = (µ+iλ)y of scalar convection diffusion
systems, this study examines the stability limits, dispersion and dissipation errors
of four diagonally implicit Runge-Kutta-type schemes on the complex (µ+iλ)∆t
plane. 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 computations
may not always remain stable, due to the inherent dispersion and dissipation
errors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate the
convergence rate, and the dispersion errors may cause large oscillations of the numerical
solutions. These errors, especially those of high wavenumber components, grow
at 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 time
step increases. It is concluded that an optimal implicit time integration scheme for
steady state simulations should have high dissipation and low dispersion.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2016-0032},
url = {https://global-sci.com/article/80206/on-the-stability-and-cpu-time-of-the-implicit-runge-kutta-schemes-for-steady-state-simulations}
}