The next question
A Fourier derivative is easy once a mode has been identified. What happens when the equation multiplies fields together and creates frequencies that were absent from the initial condition? And how do you know a fast FFT program has solved the problem you meant to give it?
We will study periodic Allen–Cahn dynamics on \([0,2\pi)\):
\[u_t=D u_{xx}+u-u^3,\qquad D=0.1,\] \[u(x,0)=0.2+0.35\cos x+0.15\cos(3x).\]
This equation combines linear diffusion with a nonlinear reaction that favours the uniform states \(u=\pm1\). Unlike the diffusion course, the mean generally changes. We will follow the same initial field through Fourier representation, nonlinear products, time integration and verification.
Six chapters, one runnable model
| Chapter | Question you will answer |
|---|---|
| 1. Fourier representation | Where does normalization belong, and which coefficients does a real FFT store? |
| 2. Nonlinear products | Why does a cubic need a different dealiasing bound from a quadratic? |
| 3. Exponential time stepping | What does ETD integrate exactly, and where does time error remain? |
| 4. OpenPFC implementation | How do physics, projection, FFTs and ETD2 fit together in real C++? |
| 5. Verification | Can a direct coefficient calculation independently check the FFT solver? |
| 6. Distributed transforms | Which rank owns each physical cell and Fourier coefficient? |
Allow roughly three hours to work through the derivations, experiments and code; take longer for the exercises. The browser calculations run locally and offer parameter links and CSV downloads. The implementation chapter runs an actual HeFFTe/FFTW-backed CPU model. Recorded model output is explicitly distinguished from browser reference calculations.
Prerequisites and scope
Start with the sampling, derivative and stability lessons if Fourier modes or amplification factors are new. You will use complex exponentials, elementary ODEs and basic C++. For the program, install an OpenPFC spectral CPU SDK, including HeFFTe/FFTW and its matching MPI/compiler stack. A GPU is not needed.
Our retained representation is \(u_K=\sum_{k=-K}^K c_k e^{ikx}\). The runnable example embeds this 1D problem in an \(N\times4\times4\) periodic FFT grid, retaining only modes \((k,0,0)\). With \(4K<N\), its cubic product is alias-free in the retained band. Its ETD2 time update still has time-discretization error, and finite \(K\) still has spatial truncation error.
This is a course in periodic Fourier methods. Nonperiodic boundaries, Chebyshev methods, general 3D pattern formation, GPU qualification and performance scaling need their own treatment. Passing these checks does not establish those properties.