Unconditional stability for multistep imex schemes: Theory

Rodolfo R. Rosales, Benjamin Seibold, David Shirokoff, Dong Zhou

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


This paper presents a new class of high order linear ImEx (implicit-explicit) multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed ImEx matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.

Original languageEnglish (US)
Pages (from-to)2336-2360
Number of pages25
JournalSIAM Journal on Numerical Analysis
Issue number5
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


  • High order time stepping
  • ImEx stability
  • Linear multistep ImEx
  • Unconditional stability


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