Numerical solution of partial differential equations using the discrete Fourier transform
DOI:
https://doi.org/10.31349/RevMexFisE.22.020221Keywords:
Partial differential equations of physics; numerical methods; Fourier transformAbstract
In this paper, we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of the method arising from the Sampling Theorem, which defines the critical Nyquist frequency and the aliasing effect. We then define the Fourier Transform (FT) and the FFT in a way that can be implemented in one and more dimensions. Finally, we show how to apply the FFT in the solution of PDEs related to problems involving two spatial dimensions, specifically the Poisson equation, the diffusion equation and the wave equation for elliptic, parabolic and hyperbolic cases, respectively.
.
References
F. S. Guzmán, Numerical Methods for Initial Value Problems in Physics, (Springer Cham, Switzerland, 2023), https://doi.org/10.1007/978-3-031-33556-3
R. Becerril, F.S. Guzmán, A. Rendon-Romero, and S. Valdez-Alvarado, Solving the time-dependent schroedinger equation using finite difference methods, Rev. Mex. Fis. E 54 (2008) 120
F. S. Guzmán, Solución de la ecuación de onda como un problema de valores iniciales usando diferencias finitas, Rev. Mex. Fis. E 56 (2010) 51
I. Avilés et al., Pseudospectral vs finite differences methods in Numerical Relativity, Rev. Mex. Fis. E 63 (2017) 25
F. Lora-Clavijo, M. D. D.-Morales F. S. Guzmán, Revisiting spherically symmetric relativistic hydrodynamics, Rev. Mex. Fis. E 58 (2012) 84
F. D. Lora-Clavijo et al., Exact solution of the 1D riemann problem in Newtonian and relativistic hydrodynamics, Rev. Mex. Fis. E 59 (2013) 28
F. S. Guzmán Murillo et al., Spherical accretion of a perfect fluid onto a black hole, Rev. Mex. Fis. E 18 (2021) 020206 1- 24, https://doi.org/10.31349/RevMexFisE.18.020206
J. W. Cooley and J. W. Tukey, An Algorithm for the Machine Calculation of Complex Fourier Series, Math. Comput. 19 (1965) 297, https://doi.org/10.2307/2003354
M.R. Spiegel, Schaum’s Outline of Fourier Analysis with Applications to Boundary Value Problems, Schaum’s Outline Series (McGraw Hill LLC, 1974)
W. H. press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1986)
P. Mocz et al., Galaxy formation with BECDM I. Turbulence and relaxation of idealized haloes, Mon. Not. Roy. Astron. Soc. 471 (2017) 4559, https://doi.org/10.1093/mnras/stx1887
I. Álvarez-Rios and F. S. Guzmán, Exploration of simple scenarios involving fuzzy dark matter cores and gas at local scales, Mon. Not. Roy. Astron. Soc. 518 (2022) 3838, https://doi.org/10.1093/mnras/stac3395
W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod. 6 (2013) 1, https://doi.org/10.3934/krm.2013.6.1
F. Guzmán-Cajica and F. S. Guzmán, Variational quantum crank-nicolson and method-oflines schemes for the solution of initial value problems, Phys. Rev. A 110 (2024) 042415, https://doi.org/10.1103/PhysRevA.110.042415
O. Amaro and D. Cruz, A Living Review of Quantum Computing for Plasma Physics, (2023), https://doi.org/10.48550/arXiv.2302.00001
Y. Y. Liu et al., Application of a variational hybrid quantumclassical algorithm to heat conduction equation and analysis of time complexity, Phys. Fluids 34 (2022) 117121, https://doi.org/10.1063/5.0121778
F. Yew Leong, W.-B. Ewe, and D. Enshan Koh, Variational Quantum Evolution Equation Solver, Sci. Rep. 12 (2022) 10817, https://doi.org/10.1038/s41598-022-14906-3
A. Sarma, T. W. Watts, M. Moosa, Y. Liu, and P. L. McMahon, Quantum Variational Solving of Nonlinear and Multi-Dimensional Partial Differential Equations, Phys. Rev. A 109 (2024) 062616, https://doi.org/10.1103/PhysRevA.109.062616
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 D. E. Rodríguez-Lara, I. Álvarez-Ríos, F. S. Guzmán
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
