Spectral Methods

PDEs · Axiom Academy

High-accuracy PDE solvers using global basis functions and exponential convergence Spectral methods expand the solution as a sum of global basis functions. The two most common choices are: The animation below shows how Fourier basis functions (sine and cosine waves) combine to approximate a smooth target function. Notice how each basis function spans the entire domain. 2. Spectral Accuracy: Exponential Convergence The hallmark of spectral methods is their exponential convergence for smooth functions. As you increase the number of basis functions N, the error decreases exponentially: Compare this to finite differences (algebraic convergence): The animation below compares convergence rates. Watch how spectral methods (green) drop to machine precision with far fewer grid points than finite differences (blue). Computing derivatives in spectral space is efficient because derivatives of basis functions are known analytically. For Fourier methods: Pseudospectral (collocation) methods evaluate nonlinear terms in physical space and derivatives in spectral space: The animation shows this transform cycle: physical space → spectral space (for derivatives) → physical space (for nonlinear terms). 4. Fast Fourier Transform for Efficiency The Fast Fourier Transform (FFT) makes spectral methods computationally practical by reducing the cost of transforming between physical and spectral space:

This is the written version of the interactive lesson above. See the full PDEs course.