Multi-time-step integration

In numerical analysis, multi-time-step integration, also referred to as multiple-step or asynchronous time integration, is a numerical time-integration method that uses different time-steps or time-integrators for different parts of the problem. There are different approaches to multi-time-step integration. They are based on domain decomposition and can be classified into strong (monolithic) or weak (staggered) schemes.^[1][2][3] Using different time-steps or time-integrators in the context of a weak algorithm is rather straightforward, because the numerical solvers operate independently. However, this is not the case in a strong algorithm. In the past few years a number of research articles have addressed the development of strong multi-time-step algorithms.^[4][5][6][7] In either case, strong or weak, the numerical accuracy and stability needs to be carefully studied.^[8] Other approaches to multi-time-step integration in the context of operator splitting methods have also been developed; i.e., multi-rate GARK method and multi-step methods for molecular dynamics simulations.^[9]

References

1. Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation. Oxford University Press. 1999-07-29. ISBN 9780198501787.
2. Toselli, Andrea; Widlund, Olof B. (2005). Domain Decomposition Methods — Algorithms and Theory – Springer. Springer Series in Computational Mathematics. Vol. 34. doi:10.1007/b137868. ISBN 978-3-540-20696-5.
3. Felippa, Carlos A.; Park, K. C.; Farhat, Charbel (2001-03-02). "Partitioned analysis of coupled mechanical systems". Computer Methods in Applied Mechanics and Engineering. Advances in Computational Methods for Fluid-Structure Interaction. 190 (24–25): 3247–3270. Bibcode:2001CMAME.190.3247F. doi:10.1016/S0045-7825(00)00391-1.
4. Gravouil, Anthony; Combescure, Alain (2001-01-10). "Multi-time-step explicit–implicit method for non-linear structural dynamics" (PDF). International Journal for Numerical Methods in Engineering. 50 (1): 199–225. Bibcode:2001IJNME..50..199G. doi:10.1002/1097-0207(20010110)50:1<199::AID-NME132>3.0.CO;2-A. ISSN 1097-0207.
5. Prakash, A.; Hjelmstad, K. D. (2004-12-07). "A FETI-based multi-time-step coupling method for Newmark schemes in structural dynamics". International Journal for Numerical Methods in Engineering. 61 (13): 2183–2204. Bibcode:2004IJNME..61.2183P. doi:10.1002/nme.1136. ISSN 1097-0207.
6. Karimi, S.; Nakshatrala, K. B. (2014-09-15). "On multi-time-step monolithic coupling algorithms for elastodynamics". Journal of Computational Physics. 273: 671–705. arXiv:1305.6355. Bibcode:2014JCoPh.273..671K. doi:10.1016/j.jcp.2014.05.034. S2CID 1998262.
7. Karimi, S.; Nakshatrala, K. B. (2015-01-01). "A monolithic multi-time-step computational framework for first-order transient systems with disparate scales". Computer Methods in Applied Mechanics and Engineering. 283: 419–453. arXiv:1405.3230. Bibcode:2015CMAME.283..419K. doi:10.1016/j.cma.2014.10.003. S2CID 15850768.
8. Zafati, Eliass (January 2023). "Convergence results of a heterogeneous asynchronous newmark time integrators". ESAIM: Mathematical Modelling and Numerical Analysis. 57 (1): 243–269. doi:10.1051/m2an/2022070. eISSN 2804-7214. ISSN 2822-7840.
9. Jia, Zhidong; Leimkuhler, Ben (2006-01-01). "Geometric integrators for multiple time-scale simulation". Journal of Physics A: Mathematical and General. 39 (19): 5379. Bibcode:2006JPhA...39.5379J. doi:10.1088/0305-4470/39/19/S04. ISSN 0305-4470.