The Monge–Ampère trajectory correction for semi-Lagrangian schemes

Requiring that numerical estimates of the flow trajectories comply with the fundamental Euler expansion formula that governs the evolution of a volume of fluid leads to a second-order nonlinear Monge–Ampère partial differential equation (MAE). In Cossette and Smolarkiewicz (2011) [15], a numerical algorithm based on solving the MAE with an inexact Newton–Krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi-Lagrangian scheme in the context of an incompressible fluid. Here we extend the theoretical analysis of the elemental rotational and deformational motions presented in Cossette and Smolarkiewicz (2011) [15]. In particular, closed-form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness. Scalar advection shows that the MA correction improves mass conservation substantially by suppressing anomalous fluid contraction. The impact of the MA correction on complex flows is studied in the framework of an ideal magneto-fluid in which the formation of current sheets leads to topological changes and reconnection of field lines. The use of the MA correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy. 2D and 3D examples are presented and the computational performance of the solver is documented.