Spectral Solution with a Subtraction Method to Improve Accuracy

This work addresses the solution to a Dirichlet boundary value problem for the Poisson equation in 1-D, d2u/dx2 = f using a numerical Fourier collocation approach. The order of accuracy of this approach can be increased by modifying f so the periodic extension of the right hand side is suffciently smooth. A proof for the order is given by Sköllermo. This work introduces a subtraction technique to modify the function\u27s right hand side to reduce the discontinuities or improve the smoothness of its periodic extension. This subtraction technique consists of cosine polynomials found by using boundary derivatives. We subtract cosine polynomials to match boundary values and derivatives of f. The derivatives need only be calculated numerically and approximately represent derivatives at the boundaries. Increasing the number of cosine polynomials in the subtraction technique increases the order of accuracy of the solution. The use of cosine polynomials matches well with the Fourier transform approach and is computationally efficient. Implementation of this technique results in a solution with variable accuracy depending on the number of collocation points and approximated boundary derivatives. Results show that the technique can be up to 14th order accurate

Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergencePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pd