Non-periodic Fourier propagation algorithms for partial differential equations

TL;DR

This paper introduces a Fourier spectral method using fast sine and cosine transforms for solving PDEs with non-periodic boundaries, demonstrating higher accuracy and efficiency over traditional polynomial or finite element methods.

Contribution

The paper develops a novel Fourier spectral approach employing DST and DCT for non-periodic boundary problems, improving accuracy and computational speed for PDEs with rapidly varying solutions.

Findings

Method achieves machine-precision accuracy for 1D heat equation.

Outperforms polynomial spectral and finite element methods in accuracy.

Faster computation due to efficient fast transforms.

Abstract

Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely used, especially for slowly-varying solutions, non-uniform spatial grids can give larger spatial discretization errors if the solutions change rapidly in space. Here, we implement a Fourier method that employs fast trigonometric expansions on a uniform grid with non-periodic boundaries using fast discrete sine transforms (DST) or/and discrete cosine transforms (DCT) to solve parabolic PDEs. We implement this method in two ways: either using a Fourier spectral derivative or a Fourier interaction picture approach. These methods can treat vector fields with a combination of Dirichlet and/or Neumann boundary conditions in one or more space dimensions. We use them to solve a variety of PDEs with analytical solutions, including the Peregrine solitary wave solution. For the 1D heat equation problem, our method with an interaction picture is accurate up to the machine precision. A soluble example of an SPDE with non-periodic boundaries is also treated. We compare the results obtained from these algorithms with those from publicly available solvers that use either polynomial spectral or finite element methods. For problems with solutions that vary rapidly in space, our method outperforms the other methods by recording lower spatial discretization errors, as well being faster in many cases, due to the efficiency improvements given by fast transforms.