Fourier Spectral Methods for Nonlocal Models

Abstract

Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in Alali and Albin (Appl Anal :1-21, 2019). It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in n dimensions reduces to the computation of a 1D smooth function given in terms of 2 F 3. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are discussed.

A general and fast convolution-based method for peridynamics: Applications to elasticity and brittle fracture

“…Recall that, in general, Fourier spectral discretizations show spectral accuracy (exponential convergence rate) if the solution is smooth on 𝕋, and if the Fourier multipliers [14] (Fourier transform of the kernels of PD operator, i.e. 𝓕(𝑐 𝑛 ) in Eq.…”

“…( 90)) are computed exactly. For example, the Fourier multipliers for the PD Laplacian operator can be expressed in terms of hypergeometric functions [14,20]. While such analytical formulas for the Fourier multipliers lead to spectral accuracy, they depend on the kernels' forms and may not be always easy to find.…”

“…The natural convolutional structure of PD formulations can be exploited using Fourier transforms as recently shown in [14,15]. Using Fast Fourier Transform (FFT) algorithms, the new scaling for PD computations drops to 𝑂(𝑁log 2 𝑁) [16,17].…”

“…These methods, however, are restricted to periodic domains. Fourier spectral methods have been used for nonlocal Allen-Cahn equation [18], fractional-in space reaction diffusion [19], and peridynamic diffusion and wave operators [14,[20][21][22], all in periodic domains. Another class of 𝑂(𝑁log 2 𝑁) methods was also introduced for 1D and 2D problems in [23][24][25], but these are still restricted to simple geometries and certain horizon shapes.…”

Computation of Eigenvalues for Nonlocal Models by Spectral Methods

“…Our work is inspired by some studies of weak eigenvalues formulation for some local and nonlocal peridynamic operators (see [2,1,12]).…”