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

Axioms

Donald A. McLaren¹  Lucy J. Campbell² 

[1] Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, Ontario K1N 6N, Canada; E-Mail:;School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada; E-Mail:

关键词: wavelet;    multiscale;    partial differential equation;    Rossby wave problem;   

DOI  :  10.3390/axioms2020142

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2013 by the authors; licensee MDPI, Basel, Switzerland.