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On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations

Part of: Basic methods in fluid mechanics Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  21 July 2016

Yongle Du  and
John A. Ekaterinaris
Yongle Du*
Affiliation:
Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA
John A. Ekaterinaris*
Affiliation:
Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA
*
*Corresponding author. Email addresses:yongle.du@gmail.com (Y. Du), ekaterij@erau.edu (J. A. Ekaterinaris)
*Corresponding author. Email addresses:yongle.du@gmail.com (Y. Du), ekaterij@erau.edu (J. A. Ekaterinaris)
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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)Δt 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.

Keywords

Runge-Kuttaimplicit time integrationdispersiondissipationnumerical stabilitysteady state simulation

MSC classification

Secondary: 65M22: Solution of discretized equations 65M12: Stability and convergence of numerical methods 76M20: Finite difference methods 76M25: Other numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 2 , August 2016 , pp. 486 - 511
Copyright
Copyright © Global-Science Press 2016 

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