Simple Spectral PDEs

This small repository implements pseudospectral solvers for a few one-dimensional partial differential equations (PDEs) in periodic domains. In each case, the solver uses in-place Fourier transforms (FFTW) and time-steppers from OrdinaryDiffEq.jl for fast, high-accuracy, and stable solutions. The density of the grid can be any power of two.

Look at the short testing script for an example of how to integrate a model.

The three equations implemented are briefly described below, in order of increasing complexity. A different equation could be implemented pretty easily by following the same pattern in the module.

Advection-Diffusion

$$ u_t = -v(x,t)u_x + D u_{xx} $$

where $v(x,t)$ is a velocity function that can be any periodic function of space and time and $D$ is a diffusion coefficient. The example below shows a solution from a random initial condition on a 512 point grid with $v(x,t)=1.25 + \cos(x + 2 \pi t/25)$ and $D=10^{-4}$. The diffusion acts mostly in the migrating low-velocity zone where the ripples are compressed and $u_{xx}$ is highest.

Korteweg–De Vries

$$ u_t = -u u_x - a^2u_{xxx} $$

This is a simplified shallow water wave model with a third-order derivative and rippling, ghostly solutions.

Kuramoto-Sivashinsky

$$ u_t = -u_{xx} - u_{xxxx} - u u_{x} $$

A fourth-order PDE famous for its chaotic behavior. The model produces different behavior for different domain length, which causes the terms above to have different scaling.

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More info about pseudospectral techniques generally can be found in the books:

• Chebyshev and Fourier Spectral Methods (Boyd)
• Spectral methods: fundamentals in single domains (Canuto)
• A Practical Guide to Pseudospectral Methods (Fornberg)

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This repo uses the Julia Language and DrWatson to make a reproducible scientific project named

Simple Spectral PDEs

It is authored by Mark Baum markmbaum@protonmail.com.

To (locally) reproduce this project, do the following:

0. Download this code base. Notice that raw data are typically not included in the git-history and may need to be downloaded independently.
1. Open a Julia console and do:

   julia> using Pkg
   julia> Pkg.add("DrWatson") # install globally, for using `quickactivate`
   julia> Pkg.activate("path/to/this/project")
   julia> Pkg.instantiate()
   

This will install all necessary packages for you to be able to run the scripts and everything should work out of the box, including correctly finding local paths.