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Abstract

Evolutions in Mechanical Engineering

Multiscale Simulations of Parabolic, Elliptic and Hyperbolic, Linear PDE’s Using Wavelet-based Multiresolution Analysis

  • Open or CloseJB Allen*

    Information Technology Laboratory, US Army Engineer Research and Development Center, USA

    *Corresponding author:JB Allen, Information Technology Laboratory, US Army Engineer Research and Development Center, USA

Submission: November 15, 2024;Published: December 16, 2024

DOI: 10.31031/EME.2024.05.000621

ISSN: 2640-9690
Volume5 Issue 5

Abstract

In this work we investigate the use of wavelet-based multiresolution analysis for scaled solutions of various linear partial differential equations. In particular, we investigate exact and scaled solutions of various parabolic, elliptic and hyperbolic partial differential equations in one and two dimensions. For the exact solutions, we utilize a standard finite difference approach with Gaussian elimination, while for the scaled solutions, we apply the wavelet-based multiresolution analysis (incorporating the Haar wavelet basis and the Schur complement). In addition to proving effective at offering scaled solutions for a select number of different partial differential equation types, we also observed that for hyperbolic equations with relatively large velocity magnitudes, an observed lag is found with respect to the wavelet-based solutions as compared to the exact solutions.

Keywords:Wavelets; Multiresolution analysis; Scaled solutions; Partial differential equations; Schur complement

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