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Incoming links: Fourier spectral method
Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 40E b Solution Created 2026-09-24 Updated 2026-09-29
Expand in the two-dimensional Fourier basis. Applying the derivative and product rules from part (a), the coefficient of the equation isThe normalization and compatibility conditions areThus, on , the infinite Fourier spectral method system isFor a square spectral truncation, chooseretain equations and unknowns indexed by , and set omitted coefficients to zero. This is the Fourier–Galerkin method: it projects the residual onto the retained Fourier modes. Other finite mode sets may be used in the same way.
- Constant thermal diffusivity makes the Fourier spectral method diagonal: each Fourier mode can be evolved by its exact exponential multiplier.
- The smooth initial Gaussian function has rapidly decaying Fourier coefficients, so a moderate grid resolves it well. The Fast Fourier transform provides an efficient change between the spatial grid and those coefficients.
- A discrete Fourier transform describes a periodically repeated array. Opposite edges are identified by periodic boundary conditions; this is a periodically continued plate model, rather than a plate with prescribed cold or insulated edges.
- The initial hot spot is far from those edges. Over the stated short interval its width remains much smaller than the side length, so the periodic images have negligible influence near the centre. At sufficiently long times, periodic diffusion tends to the conserved spatial mean.
For consistency with the supplied wavenumbers, use an FFT grid on a periodic interval: , , and similarly for . In MATLAB, replace the endpoint-inclusive coordinate definitions by
x = (0:nx-1)*Lx/nx; and y = (0:ny-1)*Ly/ny;. The supplied linspace arrays instead have spacing while the Fourier multipliers assume . Including both copies of the periodic endpoint creates a grid inconsistency, separate from high-frequency aliasing.