Fourier Series Expansion

The Fourier Series method is excellent for dissipative problems DUFFY1993 and widely used in the hydrological community to perform long time integration of weekly damped systems. For these problems, it is capable of providing spectral accuracy in time similar to Fourier (FFT) spectral accuracy in space DAVIESAJ2002, POPESCU2002, SUDICKY1990.

It utilizes the standard Bromwich contour to rewrite the inverse Laplace transform integral

in the form of a Fourier transform

For real functions , such as one typically encounters in dissipative problems, this integral can be approximated using the trapezoidal rule

The result is a weighted Fourier transform and after numerical integration, a Fourier series.

As for the Post-Widder formula approach, the resulting sum converges slowly and thus requires a series acceleration method such as the quotient-difference algorithm utilized by De Hoog, Knight, and Stokes DEHOOG1982. This methods allows a power series to be translated into an equivalent continued fraction WIMP1981

Here . The advantage of this type of acceleration method is that the coefficients need only be computed once and stored. The time domain solution at is then determined from the recursion relations