1D extended Cahn-Hilliard PDE solver

Spectral code for solving the extended Cahn-Hilliard PDE in 1D. It describes zonal flow formation in plasmas & planetary atmospheres.

The Cahn-Hilliard equation [wikipedia definition] is a nonlinear partial differential equation (PDE) which describes 'layering' phenomenon, i.e. the formation of larger domains from smaller ones. When 'extended' to include a friction term, the equation describes the formation of domains of a well-defined scale. This equation is also a model for the simplified dynamics of zonal flow shear Z(x,t) associated to 'ExB zonal flow staircase' in magnetized plasmas, for a slab geometry (x is the radial direction, in the slab approximation). It can also describe zonal flows in planetary atmospheres, where the friction is due to bottom drag. The extended Cahn-Hilliard equation takes the form:

with μ the zonal flow friction/bottom drag (dissipation).

The function 'ext_ch_pde.m' solves the 1D extended Cahn-Hilliard equation, using a spectral method, and displays the result in both real and wavenumber space.

The syntax is:

[ua,va,tv]=ext_ch_pde_matlab(mu,n,ssteps,psteps)

where the input parameters are number of grid points (n), friction (mu). 'ssteps' is the number of internal time steps, and 'psteps' is the number of output time steps.

The output fields are

'ua':the Cahn-Hilliard field (here, zonal flow shear).

'va': its Fourier transform, and:

'tv': the time vector

The matlab script uses the anti-aliasing function 'aap' from the following reference:

Hannes Uecker, 'A short ad hoc introduction to spectral methods for parabolic PDE and the Navier-Stokes equations' p.15

Lecture given at the International Summer School Modern Computational Science (August 16-28, 2009, Oldenburg, Germany)

https://pde2path.uol.de/hu/pre/030-mcs-hu.pdf