Numerical computation of the half Laplacian by means of a fast convolution algorithm

TL;DR

This work develops a fast, spectrally accurate pseudospectral method to approximate the half-Laplacian $(-\Delta)^{1/2}$ on $\mathbb{R}$ by mapping to a finite interval $[0,\pi]$ and exploiting periodic Fourier representations. A key contribution is the explicit finite-sum representation for the odd-frequency modes $(-\Delta)_s^{1/2}e^{iks}$ using Gaussian hypergeometric functions ${}_2F_1$, combined with a fast convolution framework that enables evaluation for extremely large numbers of modes with $\mathcal{O}(P\log P)$ complexity. The method covers both periodic and nonperiodic data through appropriate extensions and transforms, including a nonperiodic-to-periodic reduction strategy to improve regularity and accuracy. The authors demonstrate spectral accuracy on several test functions and successfully simulate a fractional Fisher's equation, illustrating the method's capability to resolve accelerating traveling fronts in nonlocal PDEs.

Abstract

In this paper, we develop a fast and accurate pseudospectral method to approximate numerically the half Laplacian $(-Δ)^{1/2}$ of a function on $\mathbb{R}$, which is equivalent to the Hilbert transform of the derivative of the function. The main ideas are as follows. Given a twice continuously 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 $s\in[0,π]$, which maps $\mathbb{R}$ into $[0,π]$, and denote $(-Δ)_s^{1/2}u(x(s)) \equiv (-Δ)^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of $u(x(s))$, the problem is reduced to computing $(-Δ)_s^{1/2}e^{iks} \equiv (-Δ)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. On a previous work, we considered the case with $k$ even for the more general power $α/2$, with $α\in(0,2)$, so here we focus on the case with $k$ odd. More precisely, we express $(-Δ)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussian hypergeometric function ${}_2F_1$, and also as a well-conditioned finite sum. Then, we use a fast convolution result, that enable us to compute very efficiently $\sum_{l = 0}^Ma_l(-Δ)_s^{1/2}e^{i(2l+1)s}$, for extremely large values of $M$. This enables us to approximate $(-Δ)_s^{1/2}u(x(s))$ in a fast and accurate way, especially when $u(x(s))$ is not periodic of period $π$. As an application, we simulate a fractional Fisher's equation having front solutions whose speed grows exponentially.

Figures (13)

• Figure 1: Errors in the numerical approximation of $(-\Delta)^{1/2}u(x)$, for $u(x) = (1+x^4)^{-1}$, taking $N \in\{2^2, 2^3, \ldots, 2^{13}\}$ and $L\in\{0.01, 0.02, \ldots, 10\}$. Left: we have considered an even extension at $s = \pi$. Right: we have considered an odd extension at $s = \pi$.
• Figure 2: Errors in the numerical approximation of $(-\Delta)^{1/2}u(x)$, for $u(x) = (1+x^2)^{-1/2}$, taking $N \in\{2^2, 2^3, \ldots, 2^{13}\}$. Left: we have considered an even extension at $s = \pi$, and taken $L\in\{0.01, 0.02, \ldots, 400\}$. The graphic corresponding to $U(s)$ with $s\in[0,\pi]$ and no extension is identical. Right: we have considered an odd extension at $s = \pi$, and taken $L\in\{0.01, 0.02, \ldots, 10\}$.
• Figure 3: Errors in the numerical approximation of $(-\Delta)^{1/2}u(x)$, for $u(x) = (1+x^2)^{-1/2}$, considering an even extension at $s = \pi$, and taking $N \in\{2, 3, \ldots, 10^4\}$ and $L \in\{2^{-1}, 2^0, \ldots, 2^7\}$.
• Figure 4: Errors in the numerical approximation of $(-\Delta)^{1/2}u(x)$, for $u(x) = \arctan(x)$, taking $N \in\{2^2, 2^3, \ldots, 2^{13}\}$. Left: We have considered an even extension of $U(s)$ at $s = \pi$, and taken $L\in\{0.01, 0.02, \ldots, 400\}$. Right: we have considered the smooth extension given by \ref{['e:usmooth']} for $s\in[\pi,2\pi]$, and taken $L\in\{0.01, 0.02, \ldots, 10\}$.
• Figure 5: Errors in the numerical approximation of $(-\Delta)^{1/2}u(x)$, for $u(x) = x(1+x^2)^{-1/2}$, taking $L\in\{0.01, 0.02, \ldots, 10\}$ and $N \in\{2^2, 2^3, \ldots, 2^{13}\}$. We have considered an even extension of $U(s)$ at $s = \pi$.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5
• proof