Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems

TL;DR

This paper presents a semi-Lagrangian subgrid reconstruction framework for advection-dominated multiscale problems, improving upon traditional methods by incorporating local inverse problems into a stabilized basis, with promising results in 1D and 2D.

Contribution

It introduces a novel semi-Lagrangian approach for multiscale basis construction tailored to advection-dominated problems, addressing limitations of existing methods.

Findings

Successful implementation in 1D and 2D test cases

Comparison shows advantages over standard methods

Framework adaptable to other Galerkin methods and higher dimensions

Abstract

We introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the assocociated difficulties is a semi-Lagrangian based reconstruction of subgrid variablity into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a comparison to standard methods to support our ideas and discuss possible extensions to other types of Galerkin methods, higher dimensions and nonlinear problems.