A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of $\alpha$-Euler models.

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Springer
M. Griebel , M.A. Schweitzer
Lecture Notes in Computational Science and Engineering
Computational Dynamics

Frank, J.E, Gottwald, G.A, & Reich, S. (2003). A Hamiltonian Particle-Mesh Method for the Rotating Shallow Water Equations. In M Griebel & M.A Schweitzer (Eds.), Meshfree Methods for Partial Differential Equations (pp. 131–142). Springer.