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The Fourier Continuation Alternating Direction Method Applied to the Heat Equation

Chen, Stephen Elliott Wei Shin
Degree Grantor:
University of California, Santa Barbara. Mechanical Engineering
Degree Supervisor:
Frederic G. Gibou
Place of Publication:
[Santa Barbara, Calif.]
University of California, Santa Barbara
Creation Date:
Issued Date:
Engineering, Mechanical and Applied Mathematics
Fourier Continuation
Heat Equation
Alternating Direction
Dissertations, Academic and Online resources
M.S.--University of California, Santa Barbara, 2014

This thesis experiments with and tests the Fourier Continuation Alternating Direction (FC-AD) method applied to the heat equation. This method is introduced in [6-9].

The FC-AD method is a spectral method utilizing the Fourier basis. Spectral methods utilize basis functions that are nonzero over the whole domain as opposed to finite difference methods which utilize basis functions that are nonzero only in a small portion of the domain. Both have long been used to solve partial differential equations (PDEs) numerically. An advantage of spectral methods is that the error decreases exponentially for C infinity solutions.

A limitation of spectral methods, specifically those utilizing the Fourier basis, is that periodic boundary conditions are required, or the Gibbs phenomena, commonly thought of as 'ringing' near discontinuities, severely reduces the accuracy of the method near the boundaries. This requirement limits the applicability of the Fourier basis, and it is precisely this limitation that the FC-AD method attempts to alleviate.

The FC-AD method utilizes the well known Alternating Direction method developed by Peaceman and Rachford. In the case of the heat equation, the AD method splits an n dimensional PDE into a series of n ordinary differential equations (ODEs), none of which are guaranteed to have the aforementioned boundary conditions. Normally the ODEs are solved using finite difference methods. In the FC-AD method, we add a periodic extension and take the Fourier transform of the extended function, allowing us to avoid the Gibbs phenomena. This is what gives it the name 'Fourier Continuation'.

The use of the Fourier basis allows the FC-AD method to be implemented on arbitrary domains. Previously when this was attempted the truncation error near the boundaries was first order.

In Section 1 we explain how the periodic extension of a one dimensional function can be obtained, and in Section 3 we explain how to incorporate this into the Alternating Direction method. Sections 2 and 4 give bounds on the error accumulated in Sections 1 and 3 respectively, which are slightly modified versions of results first shown in.

Section 5 describes the numerical experiments performed on the FC-AD method. It begins with a replication of a series of experiments conducted in [8], in which a complicated generation term in the heat equation is used to force the solution into a predetermined form inside an arbitrary domain. We use these results to validate our implementation.

After this we move on to original experiments. Still working with the previously mentioned domain and generation term, we compare the relation between the rate of convergence of the method and the order of the extension. This is compared to the error bounds shown in Section 4 and agreement is demonstrated.

We then perform two sets of tests. The first is where we test the heat equation without a generation term and with zero Dirichlet boundary conditions using eigenfunctions to find the error results. We do this both in a circular domain and an elliptical domain with its eccentricity varied.

The second test is of the heat equation with a time dependent boundary. We find the error not from an analytic function but from high order polynomial interpolation. This is done in both a circular domain and an elliptical domain with its eccentricity varied.

The results show that the method's performance is mostly immune to the varying situations although some exceptions are noted. High order spatial convergence in arbitrary domains is observed.

Physical Description:
1 online resource (83 pages)
UCSB electronic theses and dissertations
Catalog System Number:
Inc.icon only.dark In Copyright
Copyright Holder:
Stephen Chen
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