A Semi-Lagrangian Multiscale Framework for Advection-Dominant Problems

We introduce a new parallelizable numerical multiscale method for advection-dominated problems as they often occur in engineering and geosciences. State of the art multiscale simulation methods work well in situations in which stationary and elliptic scenarios prevail but are prone to fail when the model involves dominant lower order terms which is common in applications. We suggest to overcome the associated difficulties through a reconstruction of subgrid variations into a modified basis by solving many independent (local) inverse problems that are constructed in a semi-Lagrangian step. Globally the method looks like a Eulerian method with multiscale stabilized basis. The method is extensible to other types of Galerkin methods, higher dimensions, nonlinear problems and can potentially work with real data. We provide examples inspired by tracer transport in climate systems in one and two dimensions and numerically compare our method to standard methods