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

TL;DR

The paper addresses the challenge of achieving high-order accuracy for non-periodic PDEs without incurring Gibbs oscillations or requiring edge-clustered grids. It introduces the BSPF framework, which splits a non-periodic function into a boundary-constrained B-spline part and a periodic Fourier residual, enabling near-spectral interior accuracy with FFT efficiency. The authors provide a rigorous formulation for boundary matching, derive complexity and error analyses, and demonstrate superior differentiation, integration, and PDE solving performance on Burgers’ and shallow-water equations, including mappings to non-uniform grids. The work offers a flexible, high-accuracy alternative to Chebyshev and finite-difference methods for non-periodic problems and suggests avenues for knot optimization, adaptive refinement, and scalable implementations.

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.

Figures (7)

• Figure 1: Computing the first-order derivative of a locally oscillatory function with noise: (a) the original function $f(x)$ (blue) and its non-periodic B-spline component $f_s(x)$ (orange); (b) the exact derivative (black) and the BSPF numerical result (blue); (c) error distributions to the exact $f'(x)$ by the BSPF (blue), Chebyshev (orange), and the 10th-order compact finite difference (green) methods; (d) grid convergences of the $L^\infty$ norm error of the aforementioned methods. The results of (a)-(c) are computed on the grid $N = 2000$.
• Figure 2: Computing the first-order derivative on a mapped grid: (a) analytical mapping function $\zeta(x)$ (blue) and mapped grid (purple); (b) Spatial distributions of the error magnitude of the BSPF method to the exact solution on the original uniform grid (blue) and mapped grid (purple); (c) grid convergences of the $L^\infty$ norm error of the BSPF on the uniform grid (blue) and on the mapped grid (purple). The result of (b) is computed on the grid $N = 800$.
• Figure 3: Execution time of derivative computations using the BSPF (blue) and Chebyshev (orange) methods vs. the grid number $N$. The execution time is measured by averaging the wall time of 100 runs on an AMD EPYC 7453 server CPU.
• Figure 4: Solving a 1D ODE problem through direct integration: (a) the original function $f'(x)$ (blue) and its non-periodic B-spline component $f'_s(x)$ (orange); (b) comparisons of the exact (black) and computed $f(x)$ (blue); (c) distribution of the absolute error to the exact solution; (d) grid convergence of the $L^2$ norm error of the BSPF (blue), Chebyshev (orange) and the forth-order Simpson methods (green).
• Figure 5: Solving 1D Burgers' equation with the BSPF method: (a) the traveling wave solution; (b) distribution of the absolute error in the $x-t$ phase space.