Numerical resonances

Number theory and numerical analysis of PDE: This is the evolution of the Fourier coefficients (the actions) of a particular small solution of the resonant cubic nonlinear Schrödinger equation on the 1D torus. On the right, we observe a perfect preservation of the actions, as predicted by the theory (resonant normal form). On the left, this is a simulation with the same smooth initial data, and the same numerical scheme: a splitting algorithm with CFL number of order 0.22 - enough to ensure the existence of a modified energy over a very long time.
The difference between the two pictures? Only the number of grid points in the spatial discretization: on the left K = 30 and on the right, K = 31. The main difference is that 31 is a prime number, and 30 = 2x3x5 is not...

This numerical resonance comes from the aliasing phenomenon and the resonance module of the discrete system. It can be understood by using a combination of normal form technics and backward error analysis for PDE.
For complete details, see the notes: ETH Lecture on mathematics

Resonant stepsize: Again simulations of the cubic nonlinear Schrödinger equation with small initial data. On the left we observe a nice preservation of the actions over long time (in logarithmic scale), as predicted by the theory (Birkhoff normal forms).
On the right: The same simulation, but with time step very close (0.1%) to the one used in the left figure... Here the numerical resonance is due to the non existence of the modified energy for this specific stepsize.

For more details, see Normal form and geometric numerical integration of Hamiltonian PDE. Part I: Linear equations, and Part II: Non linear equations (talk at AMSS, Chinese Academy of Sciences, May 2009):

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