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Improving Variable-Resolution Finite-Element Semi-Lagrangian Integration Schemes by Pseudostaggering

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  • 1 Recherche en Prévision Numérique, Service de l'Environnement Atmosphérique, Dorval, P.Q., Canada
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Abstract

It is known that straightforward finite-difference and finite-element discretizations of the shallow-water equations, in their primitive (uv) form, can lead to energy propagation in the wrong direction for the small scales. Two solutions to this problem have been proposed in the past. The first of these is to define the dependent variables on grids which are staggered with respect to one another, and the second is to use the governing equations in their differentiated (vorticity-divergence) form.

We propose a new scheme that works with the primitive form of the equations, uses an unstaggered grid but doesn't propagate small-scale energy in the wrong direction, works well with variable resolution, and is as computationally efficient as staggered formulations using the Primitive form of the equations. We refer to this approach as pseudostaggering since it achieve the benefits of a staggered formulation without a staggered placement of variables.

The proposed method has been tested using the two-time-level variable resolution finite-element semi-Lagrangian model of the shallow-water equations proposed by Temperton & Staniforth (1987). Our new pseudostaggered scheme yields high accuracy with time steps as long as three hours; the rms height and wind differences are smaller than or comparable to those of the Temperton and Staniforth scheme as well as to those of its semi-implicit Eulerian analogue with a much smaller time step. It leads to a 20% reduction in computational cost of the very efficient two-time-level semi-Lagrangian Temperton & Staniforth algorithm, and is an order-of-magnitude faster than its semi-implicit Eulerian analogue.

Abstract

It is known that straightforward finite-difference and finite-element discretizations of the shallow-water equations, in their primitive (uv) form, can lead to energy propagation in the wrong direction for the small scales. Two solutions to this problem have been proposed in the past. The first of these is to define the dependent variables on grids which are staggered with respect to one another, and the second is to use the governing equations in their differentiated (vorticity-divergence) form.

We propose a new scheme that works with the primitive form of the equations, uses an unstaggered grid but doesn't propagate small-scale energy in the wrong direction, works well with variable resolution, and is as computationally efficient as staggered formulations using the Primitive form of the equations. We refer to this approach as pseudostaggering since it achieve the benefits of a staggered formulation without a staggered placement of variables.

The proposed method has been tested using the two-time-level variable resolution finite-element semi-Lagrangian model of the shallow-water equations proposed by Temperton & Staniforth (1987). Our new pseudostaggered scheme yields high accuracy with time steps as long as three hours; the rms height and wind differences are smaller than or comparable to those of the Temperton and Staniforth scheme as well as to those of its semi-implicit Eulerian analogue with a much smaller time step. It leads to a 20% reduction in computational cost of the very efficient two-time-level semi-Lagrangian Temperton & Staniforth algorithm, and is an order-of-magnitude faster than its semi-implicit Eulerian analogue.

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