Abstract

We are concerned with the development of high-order space and time numerical methods based on multistep time integrators for hyperbolic systems with relaxation and optimal control problems. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. First, we will present IMEX linear multistep methods, which are able to handle all the different scales and capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Secondly, we will focus on the properties of multi-step schemes for time discretization of adjoint equations arising in optimal control problems, in particular when the constrain corresponds to hyperbolic relaxation systems and kinetic equations. Different numerical examples will confirm the theoretical analysis.