A fast distributed data-assimilation algorithm for divergence-free advection

In this paper, we introduce a new, fast data assimilation algorithm for a 2D linear advection equation with divergence-free coefficients. We first apply the nodal discontinuous Galerkin (DG) method to discretize the advection equation and then employ a set of interconnected minimax state estimators (filters) which run in parallel on spatial elements possessing observations. The filters are interconnected by means of numerical Lax-Friedrichs fluxes. Each filter is discretized in time by a symplectic Mobius time integrator which preserves all quadratic invariants of the estimation error dynamics. The cost of the proposed algorithm scales linearly with the number of elements. Examples are presented using both synthetic and real data. In the latter case, satellite images are assimilated into a 2D model representing the motion of clouds across the surface of the Earth.