aswheaton / partial-diff-eq

Numerical solutions to partial differential equations (PDEs) in discretized space and time for initial value and boundary value problems.

partial-diff-eq

Numerical solutions to partial differential equations (PDEs) in discretised space and time for initial value and boundary value problems. Contains two classes, CH_Lattice and Poisson_Lattice, and one test script.

Cahn-Hilliard Lattice

Simulates a water/oil mixture on a lattice. Uses the fast Fourier transform (FFT) to perform Laplacian and gradient calculations in parallel, and step the simulation forward using the Euler algorithm.

Poisson Lattice

Simulates a water/oil mixture on a lattice. Uses either the Jacobi or the Gauss-Seidel algorithm to step the simulation forwards, and then sets a Dirchlecht boundary condition. The FFT is trivially utilised to step the Jacobi algorithm forward in time. A three-dimensional ''checkerboard'' pattern is used to update the lattice using the Gauss-Seidel algorithm, so that adjacent sites are updated using the FFT alternatingly.

Usage

The test script pde_test.py illustrates the utility of these classes. It is invoked with:

python3 pde_test.py [l] [m] [n] [phi_0] [max_iter] [mode]

The system arguments:

l, m, n: (int) the dimensions of the lattice. n parameter is not used by Cahn-Hilliard class instances but a ''dummy'' parameter may be supplied. 50x50x50 is sufficiently large to illustrate the behaviour of both classes.

phi_0: (float) intital scalar value with which to initalise the lattice. Small random noise is added to this value on construction of an instance.

max_iter: (int) the maximum number of iterations to which the simulation will be run before exiting. Simulations on the Poisson lattice with a $-0.5 &lt;= \phi_0 &lt;= 0.5$ will converge within < 10,000 iterations. It takes approximately 1,000,000 steps for the free energy of the Cahn-Hilliard mixture to reach its equilibrium value.

mode: (str) invokes one of several test cases for the simulation (see descriptions below). Can be oildrop, monopole, wires, or SOR.

Modes

oildrop: initalises and runs an animation of the Cahn-Hilliard oil/water mixture. Values of $\phi_0 = \pm 0.5$ will demonstrate oil drop behaviour. Records and plots the free energy of the simulation as a function of iterations.

monopole: initialises a Poisson_Lattice instance and solves the Poisson equation using the Gauss-Seidel algorithm with over-relaxation for a single charge at the center of the lattice. Once converged to the solution, plots the a slice through the electric potential in the xy-plane and the resulting electric field.

wires: reappropriates the Poisson_Lattice to solve one of Maxwell's equation, with some small changes, using the Gauss-Seidel algorithm with over- relaxation for a current density along the z-axis. Plots the magnetic vector potential $A$ and the resulting magnetic field $B$.

SOR: tests a range of values for the over-relaxation parameter $omega$ and finds the optimal value for convergence, for an electric monopole on the Poisson_Lattice. Plots the iterations to convergence for each value of omega.