  Developing robust simulation tools for problems involving multiple
mathematical scales has been a subject of great interest in computational
mathematics and engineering. A desirable feature to have in a numerical
formulation for multiscale transient problems is to be able to employ different
time-steps (multi-time-step coupling), and different time integrators and
different numerical formulations (mixed methods) in different regions of the
computational domain. We present two new monolithic multi-time-step mixed
coupling methods for first-order transient systems. We shall employ unsteady
advection-diffusion-reaction equation with linear decay as the model problem,
which offers several unique challenges in terms of non-self-adjoint spatial
operator and rich features in the solutions. We shall employ the dual Schur
domain decomposition technique to handle the decomposition of domain into
subdomains. Two different methods of enforcing compatibility along the
subdomain interface will be used in the time discrete setting. A systematic
theoretical analysis (which includes numerical stability, influence of
perturbations, bounds on drift along the subdomain interface) will be
performed. The first coupling method ensures that there is no drift along the
subdomain interface but does not facilitate explicit/implicit coupling. The
second coupling method allows explicit/implicit coupling with controlled (but
non-zero) drift in the solution along the subdomain interface. Several
canonical problems will be solved to numerically verify the theoretical
predictions, and to illustrate the overall performance of the proposed coupling
methods. Finally, we shall illustrate the robustness of the proposed coupling
methods using a multi-time-step transient simulation of a fast bimolecular
advective-diffusive-reactive system.
