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Advection without Compounding Errors through Flow Map Composition

Kulkarni, C.S. and P.F.J. Lermusiaux, 2019. Advection without Compounding Errors through Flow Map Composition. Journal of Computational Physics, sub-judice.

We propose a novel numerical methodology to compute the advective transport of tracer quantities through flow map composition. This method yields numerical solutions almost devoid of compounding numerical errors, while allowing for direct parallelization in the temporal direction. The tracer advection is computed by implicitly solving the characteristic evolution through a modified transport partial differential equation and domain decomposition in the temporal direction, followed by composition with the known initial condition. The methodology allows a rigorous computation of the spatial and temporal error bounds, yields an accuracy comparable to that of Lagrangian methods, and maintains the advantages of Eulerian schemes. We further show that there exists an optimal value of the timestep which yields the minimum total numerical error in the computations, and derive the expression for this value. We develop schemes for the addition of tracer diffusion and source terms, and the implementation of boundary conditions. Finally, the methodology is applied in three examples, namely an analytical swirl flow, an idealized flow exiting a strait undergoing sudden expansion, and a realistic ocean flow in the Bismarck sea. These examples highlight the theoretical properties of the methodology as well as its efficiency, minimal numerical errors, and applicability in realistic simulations.