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

Donald Mclaren,Lucy Campbell,Rémi Vaillancourt

doi:10.3390/axioms2020142

Donald Mclaren, Lucy Campbell + Show 1 more

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https://doi.org/10.3390/axioms2020142

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Journal: Axioms	Publication Date: Apr 23, 2013
Citations: 18	License type: CC BY 4.0

Affiliation: University of Ottawa, Carleton University

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.

Highlights

Wavelets are a versatile tool for representing and analyzing functions
Wavelet transforms can be continuous, though we will focus on the discrete form
Wavelets can be used directly to efficiently approximate the solution of partial differential equations, as in [7,8]

Summary

Introduction

Wavelets are a versatile tool for representing and analyzing functions. The discrete transform decomposes a function via families of wavelets and wavelet-related basis. Wavelets can be used directly to efficiently approximate the solution of partial differential equations, as in [7,8] (where wavelet decompositions are used in every spatial direction, as well as for the time discretization). The decomposition can be used indirectly, as a tool to analyze a function and determine where greater resolution is necessary, like in [9,10]. These are not always equal to translates and scalings of a “mother” function, and so can be (more ) made compatible with, for instance, the domain of a partial differential equation. We will keep our model simple, and so use biorthogonal wavelets (symmetric, with finite support), with a few small modifications to satisfy any required boundary conditions

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