Volume 20, Issue 2
On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations

Yongle Du & John A. Ekaterinaris

Commun. Comput. Phys., 20 (2016), pp. 486-511.

Published online: 2018-04

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  • Abstract

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.

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@Article{CiCP-20-486, author = {}, title = {On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {20}, number = {2}, pages = {486--511}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0032}, url = {http://global-sci.org/intro/article_detail/cicp/11161.html} }
TY - JOUR T1 - On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations JO - Communications in Computational Physics VL - 2 SP - 486 EP - 511 PY - 2018 DA - 2018/04 SN - 20 DO - http://doi.org/10.4208/cicp.OA-2016-0032 UR - https://global-sci.org/intro/article_detail/cicp/11161.html KW - AB -

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.

Yongle Du & John A. Ekaterinaris. (2020). On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations. Communications in Computational Physics. 20 (2). 486-511. doi:10.4208/cicp.OA-2016-0032
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