B-spline periodization of Fourier pseudo-spectral method for non-periodic problems

TL;DR

The paper introduces a B-spline-periodized Fourier (BSPF) method that extends spectral accuracy to non-periodic problems, effectively handling boundary conditions and sharp gradients in PDEs.

Contribution

It develops a novel BSPF method combining B-spline approximation with Fourier residual correction for high-order accuracy in non-periodic PDEs.

Findings

BSPF outperforms Chebyshev and finite-difference schemes in accuracy.

The method effectively resolves sharp gradients and nonlinear waves.

Numerical tests confirm spectral-like convergence and boundary accuracy.

Abstract

Spectral methods are renowned for their high accuracy and efficiency in solving partial differential equations. The Fourier pseudo-spectral method is limited to periodic domains and suffers from Gibbs oscillations in non-periodic problems. The Chebyshev method mitigates this issue but requires edge-clustered grids, which does not match the characteristics of many physical problems. To overcome these restrictions, we propose a B-spline-periodized Fourier (BSPF) method that extends to non-periodic problems while retaining spectral-like accuracy and efficiency. The method combines a B-spline approximation with a Fourier-based residual correction. The B-spline component enforces the smooth matching of boundary values and derivatives, while the periodic residual is efficiently treated by Fourier differentiation/integration. This construction preserves spectral convergence within the domain and algebraic convergence at the boundaries. Numerical tests on differentiation and integration confirm the accuracy of the BSPF method superior to Chebyshev and finite-difference schemes for interior-oscillatory data. Analytical mapping further extends BSPF to non-uniform meshes, which enables selective grid refinement in regions of sharp variation. Applications of the BSPF method to the one-dimensional Burgers' equation and two-dimensional shallow water equations demonstrate accurate resolution of sharp gradients and nonlinear wave propagation, proving it as a flexible and efficient framework for solving non-periodic PDEs with high-order accuracy.