Non-periodic Fourier propagation algorithms for partial differential equations

Spectral methods for partial differential equations (PDEs) with non-periodic boundary conditions arising in computational physics often use polynomial expansions on non-uniform grids. 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. Both methods can treat vector fields with a combination of Dirichlet and/or Neumann boundary conditions in one or more space dimensions. As examples, we use them to solve a variety of computational physics 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 machine precision. Soluble examples of stochastic partial differential equation (SPDE) with non-periodic boundaries in one and two space dimensions, with physics and interdisciplinary applications are also treated. We compare the results obtained from these algorithms with publicly available solvers that use polynomial spectral methods, and study their relative performance and error scaling. Polynomial methods with non-uniform spatial grids have lower spatial discretization errors when the solutions change slowly in space, typically with large spatial grids. For problems with rapid spatial variation, Fourier methods can outperform polynomial expansions, owing to their smaller maximum space interval, and are generally faster due to the computational efficiency of discrete Fourier transform methods. We verified this by making a complexity analysis in which we studied the total error at the optimum combination of time and space steps for a given resource use.