Numerical Methods

Orthogonal Collocation Revisited

On these pages I am constructing an Ebook/Tutorial on the Orthogonal Collocation method.  I will develop the pages as time permits, so that anyone interested in the method can follow along.  The text will be done as PDF files to make it easy to print out with full formatting.  Downloadable code will be included in the form of: Fortran 90, C++, and Excel.  I would like to include Matlab, but for a free enterprise such as this, I can not justify paying the high commercial rates to acquire the software.  If someone would like to replicate these examples in Matlab, I would be happy to include them in the download section. 

 

I am calling this Orthogonal Collocation Revisited, because after having been involved during the early days of orthogonal collocation [Young and Finlayson, 1973, 1976], I had reason to use it again recently.  A review of the last 35 years revealed that some aspects of the method appear to have never been explained clearly or never explained at all.  This realization prompted this publication.

 

Here I first lay the fundamental framework for the method.  Then, I show what works and what does not work using a series of examples.  In many cases the examples are solved not only with orthogonal collocation, but also with other methods for comparison, e.g. Galerkin, moments and finite differences.  The examples also cover a variety of problems: boundary value problems, parabolic equations, hyperbolic equations, one and two spatial dimensions.  The examples show the advantages and disadvantages of the method for each type of problem.

 

We look not only at global methods, where the solution is approximated by a single polynomial, but also finite element methods.  With finite elements, polynomials are pieced together to approximate the solution.  The following serves as both an outline and the links for the project.  To cover this material to the depth I would like is a daunting task, so unless I get encouragement through emails and page hits, the scope will likely be limited

 

At this date (Mar 2011), I am putting this out on the web for the first time.  I am still missing reference and nomenclature sections, but hopefully that will not be important for a tutorial like this.  As I update and add to the project, I will try to include dates or a revision history to make it easy to see what to update.  The only codes that are complete are those for the first example.  I am still working on the others.  Hopefully, this will be enough to get started.

 

The outline/links for the project are listed below, with the completion date or last update for each segment noted:

 A more detailed outline is included in the PDF version of the introduction.

 

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since March 21, 2011