Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

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 associated difficulties is a semi-Lagrangian based reconstruction of subgrid variability 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.

Automated summary

A new framework of numerical multiscale methods is introduced for advection-dominated problems in climate sciences, addressing difficulties faced by current methods when lower order terms are dominant. The method involves a semi-Lagrangian based reconstruction of subgrid variability into a multiscale basis by solving local inverse problems, resembling a Eulerian method with multiscale stabilized basis globally. Example runs in one and two dimensions are shown, along with comparisons to standard methods to support the ideas presented. Future extensions to other types of Galerkin methods, higher dimensions and nonlinear problems are discussed.