Computational methods of associated Legendre functions

A paper on the computation methods of associated Legendre functions by Dr T. Enomoto has been published in SOLA.


A computer program called atmospheric general circulation model (AGCM) is used to simulate global atmosphere of the Earth. The laws that govern the terrestrial atmosphere are expressed in partial differential equations. These equations need to be discretized in space and time in the computer program. For the horizontal directions, discretization can be achieved by distributing grid points on the sphere and also by expressing the flow using waves of different spatial scales. Spherical harmonics are used for the waves on the sphere and expressed as the product of trigonometric functions and associated Legendre functions (ALF).

Rapid increase of computing power allows us to use ALF of high degree and order. For example, 1279 harmonics are used to obtain a horizontal resolution of 10 km. When ALF’s of truncation wave number over 1700 are computed with a three-point recurrence formula, the values are not accurate enough in double precision and the transforms between physical and spectral spaces fail. Dr Enomoto addressed this problem by using a four-point recurrence formula (Enomoto et al. 2008). Alternatively, the three-point recurrence can be used when extended floating-point arithmetic (X-number) is employed (Fukushima 2011).

The present paper suggests a solution to the problem in the method with the four-point recurrence that arises with truncation wave number over 10000 and compares the improved method and the X-number method. The results indicate the advantages of the former in accuracy and the latter in speed. The outcomes of the present study could contribute to improve the accuracy of a high-resolution AGCM.



This work was supported by JSPS KAKENHI Grant Number 15K13417.