Results from testing and verification
Still doing errortests, and (almost by accident) ran convergence tests for the combined solver, keeping dt, dx and the number of time steps constant, only varying the conversion rate. This has been done both in 1d and 2d, and put short the stochastic model completely dominates the error, making the convergence 0.5.
Fixed and added some stuff in 1d
Did some longer simulations to check the error. The results are illustrated in the figure below. Did some more tests as well, and implemented a Backward Euler solver in 1d which lets us do some more interresting tests.
Worked through most of the 1d case
Rewrote the thesis in pdf to a more structured form. Did quite thorough error testing (and found some bugs). By now equations on the from DtU = ∇D(x)∇U -∇vU +f(x,t) can be solved in 1dim and combined with random walk in some areas.Investigated the effect of adding walkers as shown in this figure
Derived the equivalent diffusion equation for random walks with drift, still working on the actual conversion wrt velocity parameter. Some errortesting is done, but the results are quite bad as we see here.
Details in the pdf.
Updated run-script and error calculations
There is still some problems in the script with regards to the error. Seems to be calculating same error several times. Still, the interface is smoother and, more importantly, we can now run the program for several conversion-rates.
thesis in pdf
New experiment Fri Oct 11 11:40:04 2013.
Starting to investigate the effects of changing the number of walkers (and by extension the conversion factor), the walk-algorithm and whether or not one should reset walkers for each PDE-timestep (there should be a better relation here).
Seems that we can force the error to be of order dt by setting Hc = a/dt where a = 100. The parameter a is very important!
New experiment Tue Oct 8 09:38:07 2013.
Same experiment but with twice the number of walkers (M=100).
New experiment Mon Oct 7 08:45:52 2013.
Testing effect of doinusing last timestep as opposed to using current timestep as input to random walk
Started with some error-analysis oct. 3 2013
Did a simulation investigating the effects of doing a single step with the random walk model. Results were a bit bad; error of roughly 10dt.. Possible improvements in number of walkers, algorithm of walk, better conversion factor wrt step-length etc.New experiment Thu Oct 3 14:18:32 2013.
Testing the effect of taking just one step with random walks.
New experiment Mon Sep 30 13:36:31 2013.
New experiment Mon Sep 30 11:56:53 2013.
Using average of PDE and Random walk solution, seems to work better than first attempt at least squares...
New experiment.
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