عنوان مقاله [English]
Interpolation is one of the essential tools for meteorological research. The most important meteorological application of interpolation is in semi-Lagrangian methods. In semi-Lagrangian methods, the departure points of the particles arriving at the regular grid are calculated and the scalar quantity is interpolated at the departure points. A common problem is the creation of spurious fluctuations in interpolation values in areas with strong gradients if polynomial interpolation of higher than first degree is used. In order to remove spurious fluctuation, it is necessary to use monotone interpolation. The concept of monotone interpolation requires that the value of the interpolation of the function at the interval between two neighboring grid points should not be greater or less than the maximum and minimum of the function at that interval, respectively. That is, no relative extrema should be produced in the interval.
The most famous method for monotone interpolation results from changing the derivatives used in cubic Hermit polynomial with respect to the slop of the function within the interpolation interval. By changing the derivatives, one can adjust the curvature of the interpolator between the beginning and the end of the interpolated interval.
The main idea of the change in derivatives is that the derivatives in each interval should not exceed by roughly three times the slop of the function on that interval. In this paper, the curvature of the cubic Hermit polynomial is adjusted between the maximum value still giving monotonicity and the minimum value, corresponding to linear interpolation between the two points. The Hermit functions need derivatives of the function on the grid points for interpolation. For this reason, the cubic Hermit was chosen for implementation of the monotonicity procedure. The monotone piecewise cubic interpolants are easy to use, of sufficient accuracy and have been widely used. This interpolation is of second order accuracy near strict local extrema due to the application of the monotonicity constraint. It is generally the case that most monotonicity-preserving methods sacrifice accuracy in order to obtain monotonicity.
Numerical experiments with this method show that the maximum curvature still giving monotonicity may bring the cubic Hermit function close to the relative maximum and minimum in the interval. The (weakly monotone) interpolant with minimum curvature between the two neighboring points may decrease the accuracy of the interpolation as the curve connecting the two points tends to a line. The best choice for curvature in terms of both accuracy and monotonicity is intermediate between the discussed maximum and minimum values.
For an important example of practical application in meteorology, the presented method is used in transforming meteorological data in horizontal and vertical directions. An error analysis suggests that interpolation in the vertical direction for transferring data from pressure to the hybrid σ - θ, exhibits a large sensitivity to the amount of curvature used in the implementation. Maintaining maximum curvature in the interpolation function reduces the distance between vertical surfaces such that the crossing of vertical surfaces will result in error.