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A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa,Ontario K1N 6N, Canada
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
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
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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.
McLaren DA, Campbell LJ, Vaillancourt R. A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations. Axioms. 2013; 2(2):142-181.
McLaren, Donald A.; Campbell, Lucy J.; Vaillancourt, Rémi. 2013. "A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations." Axioms 2, no. 2: 142-181.