Multiscale methods for advection-diffusion problems

The development of numerical methods for multiscale advection-diffusion problems presents a number of challenges. The one-scale structures may significantly in uence the coarser properties of the system, but are often impossible to solve in full details. The time integration of the evolution system is still due to the diffusion term and its stability properties have to be taken into account for its resolution. We discuss in this paper an algorithm, which combines Heterogeneous Multiscale Methods (HMM) with Orthogonal Runge-Kutta Chebyshev (ROCK) methods, for the efficient numerical resolution of multiscale advection-diffusion problems.