Divergence-free Wavelets for Navier-Stokes

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.

Figures (17)

• Figure 1: From left to right: the scaling function $\phi$ with its associated symmetric wavelet with shortest support, and their duals: the dual scaling function $\phi^*$ and the dual wavelet $\psi^*$.
• Figure 2: Scaling functions and associated even and odd wavelets with shortest support, for splines of degree 1 (left) and 2 (right).
• Figure 3: Anisotropic 2D wavelet transform.
• Figure 4: Isotropic 2D wavelet transform.
• Figure 5: Isotropic 2D generating divergence-free wavelets $\Psi_{\textrm{div}}^{(1,0)}$ (left), $\Psi_{\textrm{div}}^{(0,1)}$ (center) and $\Psi_{\textrm{div}}^{(1,1)}$ (right).