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We consider the development of high order space and time numerical methodsbased on Implicit-Explicit (IMEX) multistep time integrators for hyperbolicsystems with relaxation. More specifically, we consider hyperbolic balance lawsin which the convection and the source term may have very different time andspace scales. As a consequence the nature of the asymptotic limit changescompletely, passing from a hyperbolic to a parabolic system. From thecomputational point of view, standard numerical methods designed for thefluid-dynamic scaling of hyperbolic systems with relaxation present severaldrawbacks and typically lose efficiency in describing the parabolic limitregime. In this work, in the context of Implicit-Explicit linear multistepmethods we construct high order space-time discretizations which are able tohandle all the different scales and to capture the correct asymptotic behavior,independently from its nature, without time step restrictions imposed by thefast scales. Several numerical examples confirm the theoretical analysis.

Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation