14 - Spectral Methods

Summary

Introduction

For solving the governing equations of atmospheric motion that give rise to a coupled set of nonlinear partial differential equations, the emphasis for a solution procedure was restricted to the method of finite differences. The method of finite differences is quite popular and is still widely employed. However, as mentioned in the earlier chapters, despite its popularity, it has a number of problems associated with truncation error, linear and nonlinear instability, not to mention, the associated amplitude and phase errors of finite difference schemes. Hence, continued efforts have been initiated to seek and find a more accurate and much improved alternate solution procedure that does not have the limitations of the method of finite differences. The Galerkin method (or series expansion method) is one such solution procedure that provides for a more accurate solution to the governing equations of atmospheric motions as manifested in a coupled system of nonlinear partial differential equations.

Series Expansion Method

Series expansion method (or the Galerkin method) is a technique that provides for a more accurate solution to the governing equations of atmospheric motions. The method forms part of the general class of methods that include the two well-known methods: (i) spectral methods and (ii) finite element method. Broadly speaking, in the Galerkin method, one approximates functions as a linear combination of prescribed expansion functions, the latter known as “basis functions.” The basis functions are known to possess nice properties. For a continuous function u(x), one can write

where ϕ_j(x), j = 1,2,...,N, are the basis functions, which satisfy any boundary conditions on u(x). The coefficients uj are the unknown coefficients that form a vector of N numbers.

Consider a partial differential equation of the following form with being an ordinary differential operator, operating on u:

Substituting Equation (14.1) in Equation (14.2) leads to a mismatch between the LHS and RHS of Equation (14.2) that provides for a definition of the residual ϵ(x) given by

If the ordinary differential operator L is linear and the basis functions, ϕ_j(x) are the eigenfunctions of L, then the residual can be set to zero for the whole domain and the resulting N algebraic equations can be solved for the unknown coefficients u_j.