Multi-time-step integration explained

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]

Notes and References

  1. Book: Domain Decomposition Methods for Partial Differential Equations. 9780198501787. Oxford University Press. 1999-07-29. Numerical Mathematics and Scientific Computation.
  2. Book: Domain Decomposition Methods — Algorithms and Theory – Springer. 34. Toselli. Andrea. Widlund. Olof B.. 10.1007/b137868. Springer Series in Computational Mathematics. 2005. 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. 10.1016/S0045-7825(00)00391-1. 2001CMAME.190.3247F.
  4. Gravouil. Anthony. Combescure. Alain. 2001-01-10. Multi-time-step explicit–implicit method for non-linear structural dynamics. International Journal for Numerical Methods in Engineering. en. 50. 1. 199–225. 10.1002/1097-0207(20010110)50:1<199::AID-NME132>3.0.CO;2-A. 1097-0207. 2001IJNME..50..199G.
  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. en. 61. 13. 2183–2204. 10.1002/nme.1136. 1097-0207. 2004IJNME..61.2183P.
  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. 10.1016/j.jcp.2014.05.034. 1305.6355. 2014JCoPh.273..671K. 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. 10.1016/j.cma.2014.10.003. 1405.3230. 2015CMAME.283..419K. 15850768.
  8. Zafati . Eliass . Convergence results of a heterogeneous asynchronous newmark time integrators . ESAIM: Mathematical Modelling and Numerical Analysis . January 2023 . 57 . 1 . 243–269 . 2822-7840 . 2804-7214 . 10.1051/m2an/2022070 . free .
  9. Jia. Zhidong. Leimkuhler. Ben. 2006-01-01. Geometric integrators for multiple time-scale simulation. Journal of Physics A: Mathematical and General. en. 39. 19. 5379. 10.1088/0305-4470/39/19/S04. 0305-4470. 2006JPhA...39.5379J.

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