Robust adaptive meshing, mesh density functions, and nonlocal observations for ensemble based data assimilation

Adaptive spatial meshing has proven invaluable for the accurate, efficient computation of solutions of time dependent partial differential equations. In a DA context the use of adaptive spatial meshes addresses several factors that place increased demands on meshing; these include the location and relative importance of observations and the use of ensemble solutions. To increase the efficiency of adaptive meshes for data assimilation, robust look ahead meshes are developed that fix the same adaptive mesh for all ensemble members for the entire time interval of the forecasts and that incorporates the observations at the next analysis time. This allows for increased vectorization of the ensemble forecasts while minimizing interpolation of solutions between different meshes. The techniques to determine these robust, nonuniform meshes are based upon combining metric tensors or mesh density functions. The combination of mesh density functions is achieved through the use of so-called metric tensor intersection, a low dimensional (in the spatial dimension of the PDE) and local (pointwise in the spatial variable) technique. We illustrate the robust ensemble look ahead meshes using traveling wave solutions of a bistable reaction-diffusion equation. Observation operators based on convolution type integrals and their associated metric tensors are derived. These further the goals of making efficient use of adaptive meshes in ensemble based DA techniques, developing and employing robust meshes that are effective for a range of similar behaviors in both the ensembles and the observations, and the integration with advanced numerical PDE techniques (a quasi-Lagrangian moving mesh DG technique employing embedded pairs for time stepping). Numerical experiments with different observation scenarios are presented for a 2D inviscid Burgers’ equation, a multi-component system, a 2D Shallow Water model, and for a coupled system of two 1D Kuramoto-Sivashinsky equations.