August 1, 2015.
This report examines a new methodology in solving Partial Differential Equations (PDEs) numerically. The report also studies the accuracy of this new method as a PDE solver. This new Fourier Continuation (FC) method is one of a few that avoids the well-known Gibbs Phenomenon, which is the overestimation or underestimation of a function. These estimations are oscillations around a “jump” when a non-periodic function is expressed in terms of sines and cosines. Instead, the FC algorithm creates a smooth, periodic extension of a function over a general domain, as demonstrated by the many examples presented here. The FC algorithm was applied to a multitude of different functions, a majority of which were non-periodic; some were periodic. All functions were evaluated on at least two different domains, [0,1] and [0,10]. Some of the periodic functions were evaluated on a periodic domain. They demonstrate that FC algorithm produces a solution similar to that of the Fourier Series and Transform without the Gibbs Phenomenon. Among the functions where the FC method was applied, the errors for the data of the derivative converged at a 4th order rate. The data collected for the interpolation of the original function did not converge at all and remains inconclusive at this time. These examples indicate that the Fourier Continuation method is a viable PDE solver.
Air Force Research Laboratory (AFRL)
This material is based upon work supported by the Chevron Corporation and is made possible with contributions from the National Science Foundation under Grant No. 1340110, Howard Hughes Medical Institute, S.D. Bechtel Jr. Foundation, National Marine Sanctuary Foundation, and from the host research center. Any opinions, findings, and conclusions or recommendations expressed in this material are solely those of the authors. The STAR Program is administered by the Cal Poly Center for Excellence in STEM Education on behalf of the California State University system.