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# Eliminating the Interpolation Associated with the Semi-Lagrangian Scheme

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## Abstract

There are several reasons why it is desirable to eliminate the interpolation associated with the conventional semi-Larangian scheme. Interpolation leads to smoothing and is also the most costly operation associated with the technique. Furthermore, its elimination produces a scheme that is more readily adaptable to a spectral model.

In the conventional semi-Lagrangian method, in order to predict a field value at grid point (Xi, Yj) it is necessary to calculate the trajectory over one time step for the fluid element that arrives at (Xi, Yj). One then moves along this trajectory in order to extract the field value at an upstream location that generally lies between the grid points, and hence requires the use of interpolation formulae.

This trajectory can be represented as a vector. In the new scheme, the trajectory vector is considered to be the sum of two other vectorsâ€”a first vector joining (Xi, Yj) to the grid point (Xu, Yu) nearest the upstream location, and a second vector joining (Xu, Yu) to the upstream location. The advection along the first vector is done via a Lagrangian technique that displaces the field from one grid point to another and, therefore, does not require interpolation. The advection along the second vector is accounted for by an Eulerian approach with the advecting winds modified in such a way that the Courant number is always less than one, thus retaining the attractive stability properties of the interpolating semi-Lagrangian method.

Here the noninterpolating scheme is applied to a model of the shallow water equations and its performance is assessed by comparing the results with those produced by one model which uses the interpolating semi-Lagrangian technique, and another model which uses a fourth-order Eulerian approach. Five-day integrations indicate that the scheme is stable, accurate, and appears to have efficiency advantages.

## Abstract

There are several reasons why it is desirable to eliminate the interpolation associated with the conventional semi-Larangian scheme. Interpolation leads to smoothing and is also the most costly operation associated with the technique. Furthermore, its elimination produces a scheme that is more readily adaptable to a spectral model.

In the conventional semi-Lagrangian method, in order to predict a field value at grid point (Xi, Yj) it is necessary to calculate the trajectory over one time step for the fluid element that arrives at (Xi, Yj). One then moves along this trajectory in order to extract the field value at an upstream location that generally lies between the grid points, and hence requires the use of interpolation formulae.

This trajectory can be represented as a vector. In the new scheme, the trajectory vector is considered to be the sum of two other vectorsâ€”a first vector joining (Xi, Yj) to the grid point (Xu, Yu) nearest the upstream location, and a second vector joining (Xu, Yu) to the upstream location. The advection along the first vector is done via a Lagrangian technique that displaces the field from one grid point to another and, therefore, does not require interpolation. The advection along the second vector is accounted for by an Eulerian approach with the advecting winds modified in such a way that the Courant number is always less than one, thus retaining the attractive stability properties of the interpolating semi-Lagrangian method.

Here the noninterpolating scheme is applied to a model of the shallow water equations and its performance is assessed by comparing the results with those produced by one model which uses the interpolating semi-Lagrangian technique, and another model which uses a fourth-order Eulerian approach. Five-day integrations indicate that the scheme is stable, accurate, and appears to have efficiency advantages.

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