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.{{Cite book|url=https://global.oup.com/academic/product/domain-decomposition-methods-for-partial-differential-equations-9780198501787?cc=us&lang=en&|title=Domain Decomposition Methods for Partial Differential Equations|isbn=9780198501787|publisher=Oxford University Press|date=1999-07-29|series=Numerical Mathematics and Scientific Computation}}{{Cite book|title=Domain Decomposition Methods — Algorithms and Theory – Springer|volume = 34|last1=Toselli|first1=Andrea|last2=Widlund|first2=Olof B.|doi=10.1007/b137868|series = Springer Series in Computational Mathematics|year = 2005|isbn = 978-3-540-20696-5}}{{Cite journal|last1=Felippa|first1=Carlos A.|last2=Park|first2=K. C.|last3=Farhat|first3=Charbel|date=2001-03-02|title=Partitioned analysis of coupled mechanical systems|journal=Computer Methods in Applied Mechanics and Engineering|series=Advances in Computational Methods for Fluid-Structure Interaction|volume=190|issue=24–25|pages=3247–3270|doi=10.1016/S0045-7825(00)00391-1|bibcode=2001CMAME.190.3247F}} 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.{{Cite journal|last1=Gravouil|first1=Anthony|last2=Combescure|first2=Alain|date=2001-01-10|title=Multi-time-step explicit–implicit method for non-linear structural dynamics|url= https://hal.science/hal-04253402/file/gravouil2000.pdf|journal=International Journal for Numerical Methods in Engineering|language=en|volume=50|issue=1|pages=199–225|doi=10.1002/1097-0207(20010110)50:1<199::AID-NME132>3.0.CO;2-A|issn=1097-0207|bibcode=2001IJNME..50..199G}}{{Cite journal|last1=Prakash|first1=A.|last2=Hjelmstad|first2=K. D.|date=2004-12-07|title=A FETI-based multi-time-step coupling method for Newmark schemes in structural dynamics|journal=International Journal for Numerical Methods in Engineering|language=en|volume=61|issue=13|pages=2183–2204|doi=10.1002/nme.1136|issn=1097-0207|bibcode=2004IJNME..61.2183P}}{{Cite journal|last1=Karimi|first1=S.|last2=Nakshatrala|first2=K. B.|date=2014-09-15|title=On multi-time-step monolithic coupling algorithms for elastodynamics|journal=Journal of Computational Physics|volume=273|pages=671–705|doi=10.1016/j.jcp.2014.05.034|arxiv=1305.6355|bibcode=2014JCoPh.273..671K|s2cid=1998262}}{{Cite journal|last1=Karimi|first1=S.|last2=Nakshatrala|first2=K. B.|date=2015-01-01|title=A monolithic multi-time-step computational framework for first-order transient systems with disparate scales|journal=Computer Methods in Applied Mechanics and Engineering|volume=283|pages=419–453|doi=10.1016/j.cma.2014.10.003|arxiv=1405.3230|bibcode=2015CMAME.283..419K|s2cid=15850768}} In either case, strong or weak, the numerical accuracy and stability needs to be carefully studied.{{cite journal | last1 = Zafati | first1 = Eliass | title = Convergence results of a heterogeneous asynchronous newmark time integrators | journal = ESAIM: Mathematical Modelling and Numerical Analysis | date = January 2023 | volume = 57 | issue = 1 | pages = 243–269 | issn = 2822-7840 | eissn = 2804-7214 | doi = 10.1051/m2an/2022070 | pmid = | url = | doi-access = free }} Other approaches to multi-time-step integration in the context of operator splitting [http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture11.pdf methods] have also been developed; i.e., multi-rate GARK [http://www.imacm.uni-wuppertal.de/fileadmin/imacm/preprints/2015/imacm_15_06.pdf method] and multi-step methods for molecular dynamics simulations.{{Cite journal|last1=Jia|first1=Zhidong|last2=Leimkuhler|first2=Ben|date=2006-01-01|title=Geometric integrators for multiple time-scale simulation|journal=Journal of Physics A: Mathematical and General|language=en|volume=39|issue=19|pages=5379|doi=10.1088/0305-4470/39/19/S04|issn=0305-4470|bibcode=2006JPhA...39.5379J}}

References

Category:Numerical analysis

Category:Applied mathematics