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Axioms 2013, 2(2), 142-181; doi:10.3390/axioms2020142

A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations

1
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa,Ontario K1N 6N, Canada
2
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, OntarioK1S 5B6, Canada
*
Author to whom correspondence should be addressed.
Received: 28 February 2013 / Accepted: 8 April 2013 / Published: 23 April 2013
(This article belongs to the Special Issue Wavelets and Applications)

Abstract

This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller systems that can be solved sequentially. Included is a test on a basic non-linear problem, with both the results of the test, and the time required to calculate them, compared with control results based on a single system with fine resolution. The method is then tested on a non-trivial problem, its computational time and accuracy checked against control results. In both tests, it was found that the method requires less computational expense than the control. Furthermore, the method showed convergence towards the fine resolution control results.
Keywords: wavelet; multiscale; partial differential equation; Rossby wave problem wavelet; multiscale; partial differential equation; Rossby wave problem
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

McLaren, D.A.; Campbell, L.J.; Vaillancourt, R. A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations. Axioms 2013, 2, 142-181.

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