**Giovanni Naldi **^{[a]}

*A journey through multiscale, some episodes from approximation and modelling
*

**Pages:** 1-82

**Received:** 16 February 2017

**Accepted in revised form:** 25 July 2017

**Mathematics Subject Classification (2010):** 65M99, 35Q70, 35Q92, 65M06.

**Keywords:** Multiscale modelling, Relaxation approximation, nonlinear evolutionary
differential equations, Relaxed methods.

**Author address:**

[a]: Università degli studi di Milano, Dipartimento di Scienze e Politiche Ambientali Centro ADAMSS, via Celoria 2, Milano, 20133, Italy

**Abstract:**
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 \(\epsilon\).
When \(\epsilon \rightarrow 0^+\), the system relaxes onto the original PDE: in this way, a consistent
discretization of the relaxation system for vanishing \(\epsilon\) 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.

This research was partially supported by ADAMSS Center of the Università degli Studi di Milano.

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