Asymptotic behavior of splitting schemes involving timesubcycling techniques
Abstract
This paper deals with the numerical integration of wellposed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, timesubcycling techniques are widely used every day to improve computational efficiency. These methods rely on a decomposition of the vector field in a fast part and a slow part and take advantage of that decomposition. This way, if an unconditionnally stable (semi)implicit scheme cannot be easily implemented, one can integrate the fast equations with a much smaller time step than that of the slow equations, instead of having to integrate the whole system with a very small timestep to ensure stability. Then, one can build a numerical integrator using a standard composition method, such as a Lie or a Strang formula for example. Such methods are primarily designed to be convergent in shorttime to the solution of the original problems. However, their longtime behavior rises interesting questions, the answers to which are not very well known. In particular, when the solutions of the problems converge in time to an asymptotic equilibrium state, the question of the asymptotic accuracy of the numerical longtime limit of the schemes as well as that of the rate of convergence is certainly of interest. In this context, the asymptotic error is defined as the difference between the exact and numerical asymptotic states. The goal of this paper is to apply that kind of numerical methods based on splitting schemes with subcycling to some simple examples of evolutionary ODEs and PDEs that have attractive equilibrium states, to address the aforementioned questions of asymptotic accuracy, to perform a rigorous analysis, and to compare them with their counterparts without subcycling. Our analysis is developed on simple linear ODE and PDE toymodels and is illustrated with several numerical experiments on these toymodels as well as on more complex systems. Lie and
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.3064
 Bibcode:
 2014arXiv1410.3064D
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 IMA Journal of Numerical Analysis, Oxford University Press (OUP): Policy A  Oxford Open Option A, 2015