Divergence-free Wavelets for Navier-Stokes

TL;DR

This paper introduces divergence-free wavelets tailored for representing Navier-Stokes solutions, offering efficient algorithms and a novel method for flow decomposition, validated through numerical tests.

Contribution

It presents a new divergence-free wavelet basis and a fast algorithm for 2D and 3D flows, along with a method for Hodge decomposition of flows.

Findings

Effective representation of Navier-Stokes solutions using divergence-free wavelets

Fast algorithms developed for 2D and 3D flow computations

Numerical validation confirms the approach's accuracy

Abstract

In this paper, we investigate the use of compactly supported divergence-free wavelets for the representation of the Navier-Stokes solution. After reminding the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for 2D and 3D incompressible flows. In order to compute the nonlinear term, we propose a new method which provides in practice with the Hodge decomposition of any flow: this decomposition enables us to separate the incompressible part of the flow from its orthogonal complement, which corresponds to the gradient component of the flow. Finally we show numerical tests to validate our approach.