This is the fifth and final numerical project in FYS3150 - Computational Physics at University of Oslo, fall 2015.
In this report, I investigate the properties of the forward Euler, backward Euler and Crank-Nicolson schemes and use these methods to solve the 1+1D diffusion equation in light of diffusion of neurotransmitters in the synaptic cleft between brain cells. I also investigate the method’s stability and find that all produce results with good agreement to the analytical solution, with the poorest result being 0.4% off.
I also look at the diffusion process as a series of random walkers walking across the synaptic cleft and find that the best results are obtained when the particles are constrained to move with a constant step length, as compared to one with a normal distribution. In light of these results I propose an explanation of the underlying physics of this particular diffusion process and how the density of the diffusive medium may alter the results in a way that renders the normally distributed step lengths a more feasible approach.
C++11 with standard vector class
Python 2.7 with numpy and matplotlib
The main directory has two important files: plot.py which produces the plots for the differential methods(FE, BE and CN)
and histplot.py which produces the histograms for the random walk method. The latter also includes a test for the
random walk methods.
The subdirectories are:
Implementation the differential methods
Implementation of random walk with constant step length
Implementation of random walk with normally distributed step lengths
All files to produce the plots are stored, however, if desireable to reproduce the results, the necessary runs are:
task_d/task_d 0.1 0.1 0.1_curved
task_d/task_d 0.1 0.3 0.1_linear
task_d/task_d 0.01 0.1 0.01_curved
task_d/task_d 0.01 0.3 0.01_linear
task_f/task_f 0.1 RandomWalk_constant_step_0.100000.txt
task_f/taks_f 0.3 RandomWalk_constant_step_0.300000.txt
task_g/task_g 0.1 RandomWalk_gaussian_step_0.100000.txt
task_g/task_g 0.3 RandomWalk_gaussian_step_0.300000.txt
task_g/task_g 0.5 RandomWalk_gaussian_step_0.500000.txt