2025-10-02 –, Coffee room
Language: English
The use of Fourier series for high-order trigonometric interpolation can be found in a plethora of applications in science and engineering. Here we present a framework for constructing and testing recently-introduced extension operators that can enable such representations for non-periodic functions (while avoiding the ringing effect). This can ultimately facilitate the construction of high-performance FFT-based solvers for PDEs with general boundary conditions.
This contribution presents a package (FourierContinuation.jl) for building numerical continuation operators that enable discrete Fourier series representations of non-periodic functions while mitigating the Gibb's phenomenon. These operators can produce highly-accurate derivative approximations that are suitable for constructing high-order FFT-speed solvers for hyperbolic and parabolic PDEs governing a wide range of physical applications [1,2]. FC-based solvers have been demonstrated to incur minimal to no numerical pollution errors [1,2], making them ideal for fast and accurate long-time or long-distance simulation for scientific computing.
Many underlying methodologies for numerically constructing such operators, which are often based on least squares problems, tend to be fundamentally ill-conditioned. Hence, implementation is non-trivial, requiring high-precision or symbolic arithmetic (for this task, Julia is well-suited). Here we provide a generalized tool for researchers to use the most common approaches for producing these periodic extensions of smooth functions, and to develop/test new ones.
[1] A stable high-order FC-based methodology for hemodynamic wave propagation, F. Amlani and N.M. Pahlevan.
J. Comput. Phys. (2020)
https://doi.org/10.1016/j.jcp.2019.109130
[2] An FC-based spectral solver for elastodynamic problems in general three-dimensional domains, F. Amlani and O.P. Bruno.
J. Comput. Phys. (2016)
https://doi.org/10.1016/j.jcp.2015.11.060
