Entropic approximations of the semigeostrophic shallow water equations

We develop a discretisation of the semigeostrophic rotating shallow water equations, based upon their optimal transport formulation. This takes the form of a Moreau-Yoshida regularisation of the Wasserstein metric. Solutions of the optimal transport formulation provide the shallow water layer depth represented as a measure, which is itself the push forward of an evolving measure under the semigeostrophic coordinate transformation. First, we propose and study an entropy regularised version of the rotating shallow water equations. Second, we discretise the regularised problem by replacing both measures with weighted sums of Dirac measures, and approximate the (squared) L2 norm of the layer depth, which defines the potential energy. We propose an iterative method to solve the discrete optimisation problem relating the two measures, and analyse its convergence. The iterative method is demonstrated numerically and applied to the solution of the time-dependent shallow water problem in numerical examples.