A journey through multiscale, some episodes from approximation and modelling

The present notes contains both a survey of and some novelties about mathematical problems which emerged in multiscale based approach in approximation of evolutionary partial differential equations. Specifically, we present a relaxed systems approximation for nonlinear diffusion problems, which can tackle also the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation with a semi-linear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ϵ. When ϵ→0+, the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ϵ yields a consistent discretization of the original PDE. The advantage of this procedure is that numerical schemes obtained in this fashion do not require to solve implicit nonlinear problems and possess the robustness of upwind discretizations. We also review a unified framework, including BGK-based diffusive relaxation methods and new relaxed numerical schemes. A stability analysis for the new methods is sketched and high order extensions are provided. Finally some numerical tests in one and two dimensions are shown with preliminary results for nonlocal problems and multiscale hyperbolic systems.