Numerical computation of the half Laplacian by means of a fast convolution algorithm. ETNA - Electronic Transactions on Numerical Analysis

Résumé 0

In this paper we develop a fast and accurate pseudospectralmethod to numerically approximate the half Laplacian $(-\Delta)^{1/2}$ of afunction on $\mathbb{R}$, which is equivalent to the Hilbert transform of thederivative of the function. The main ideas are as follows. Given a twicecontinuously differentiable bounded function $u\in\mathcal C_b^2(\mathbb{R})$,we apply the change of variable $x=L\cot(s)$, with $L>0$ and $s\in[0,\pi]$,which maps $\mathbb{R}$ into $[0,\pi]$, and denote $(-\Delta)_s^{1/2}u(x(s))\equiv (-\Delta)^{1/2}u(x)$. Therefore, by performing a Fourier series expansionof $u(x(s))$, the problem is reduced to computing $(-\Delta)_s^{1/2}e^{iks}\equiv (-\Delta)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. In a previous work weconsidered the case with $k$ even for more general powers $\alpha/2$, with$\alpha\in(0,2)$, so here we focus on the case with $k$ odd. More precisely, weexpress $(-\Delta)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussianhypergeometric function ${}_2F_1$ and as a well-conditioned finite sum.Then we use a fast convolution result that enable us to compute veryefficiently $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$ for extremelylarge values of $M$. This enables us to approximate $(-\Delta)_s^{1/2}u(x(s))$in a fast and accurate way, especially when $u(x(s))$ is not periodic of period$\pi$. As an application, we simulate a fractional Fisher\'s equation havingfront solutions whose speed grows exponentially.