4
Series-Expansion Methods
Series-expansion methods that are potentially useful in geophysical fluid dynamics include the spectral method, the pseudospectral method, and the finite-element
method. The spectral method plays a particularly important role in global atmospheric models , in which the horizontal structure of the numerical solution is often
represented as a truncated series of spherical harmonics . Finite-element methods,
on the other hand, are not commonly used in multidimensional wave propagation
problems because they generally require the solution of implicit algebraic systems and are therefore not as efficient as competing explicit methods. All of these
series -expansion methods share a common foundation that will be discussed in
the next section .
4.1 Strategies for Minimizing the Residual
Suppose F is an operator involving spatial derivatives of 1/1, and that solutions are
sought to the partial differential equation
-
a.."
at + F{1/I) = 0,
(4.1)
subject to the initial condition 1/1 {x , to) = f{x) and to boundary conditions at the
edges of some spatial domain S. The basic idea in all series-expansion methods
is to approximate the spatial dependence of 1/1 as a linear combination of a finite
number of predetermined expansion functions. Let the general form of the series
D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999
Series-Expansion Methods
Series-expansion methods that are potentially useful in geophysical fluid dynamics include the spectral method, the pseudospectral method, and the finite-element
method. The spectral method plays a particularly important role in global atmospheric models , in which the horizontal structure of the numerical solution is often
represented as a truncated series of spherical harmonics . Finite-element methods,
on the other hand, are not commonly used in multidimensional wave propagation
problems because they generally require the solution of implicit algebraic systems and are therefore not as efficient as competing explicit methods. All of these
series -expansion methods share a common foundation that will be discussed in
the next section .
4.1 Strategies for Minimizing the Residual
Suppose F is an operator involving spatial derivatives of 1/1, and that solutions are
sought to the partial differential equation
-
a.."
at + F{1/I) = 0,
(4.1)
subject to the initial condition 1/1 {x , to) = f{x) and to boundary conditions at the
edges of some spatial domain S. The basic idea in all series-expansion methods
is to approximate the spatial dependence of 1/1 as a linear combination of a finite
number of predetermined expansion functions. Let the general form of the series
D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics
© Springer Science+Business Media New York 1999