Treffer: A Fast Monotone Discretization of the Rotating Shallow Water Equations.

This paper presents a new discretization of the rotating shallow water equations and a set of decisions, ranging from a simplification of the continuous equations down to the implementation level, yielding a code that is fast and accurate. Accuracy is reached by using WENO reconstructions on the mass flux and on the nonlinear Coriolis term. The results show that the implicit mixing and dissipation, provided by the discretization, allow a very good material conservation of potential vorticity and a minimal energy dissipation. Numerical experiments are presented to assess the accuracy, which include a resolution convergence, a sensitivity on the free‐slip versus no‐slip boundary conditions, a study on the separation of waves from vortical motions. Speed is achieved by a series of choices rather than a single recipe. The main choice is to discretize the covariant form written in index coordinates. This form, rooted in the discrete differential geometry, removes most of the grid scale terms from the equations, and keeps them only where they should be. The model objects appearing in resulting continuous equations have a natural correspondence with the grid cell features. The other choices are guided by the maximization of the arithmetic intensity. Finally this paper also proves that a pure Python implementation can be very fast, thanks to the possibility of having compiled Python. As a result, the code performs 2 TFlops per second using thousand cores. Plain Language Summary: Using a simplified model of the ocean and atmosphere dynamics, this paper presents a set of key decisions that yields a code that is both fast and accurate. The accuracy is assessed in terms of capacity of the code to maintain dynamic structures over long periods of time while avoiding the emergence of numerical noise in the solution. Accuracy is achieved by using a very accurate discretization on two decisive terms of the model equations. Speed is achieved by a series of choices ranging from a simplification of the continuous equations down to the implementation level. This paper also proves that a pure Python code is a viable alternative to perform simulations on high performance computing centers with as much as 2 TFlops per second using thousand cores. Key Points: Use WENO reconstructions on the mass flux and on the nonlinear Coriolis term to reach low level of energy dissipation and high accuracy on material conservation of potential vorticityExpress the continuous equations with index coordinates, finite volume quantities, covariant, and contravariant components of the velocity to minimize the number of operationsMaximize the arithmetic intensity to achieve 2 GFlops per second per core with a pure Python code [ABSTRACT FROM AUTHOR]

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