Numerical Simulations
NLS Growth: interacting recurrent packet seeds
Explore two packets in the sextic resonant model.
Compare the complete quartic-plus-sextic evolution with a control that removes mixed packet terms and with the quartic reference.
The interactive page shows carrier transport, pump actions, and full Sobolev norms on fixed Fourier sites, with model details and downloadable results.
Boltzmann Equation: DSMC
The following 2d velocity distributions have been simulated using a Direct Simulation Monte Carlo method. In particular, Bird's algorithm was applied to a initial data consisting of weighted sums of Gaussians (with variance = 1 in both directions) to solve the Boltzmann equation:
$\partial_t f(v) = \frac{1}{Kn} Q(f,f)(v), \qquad \text{ for } v \in \mathbb{R}^2$.
The collision kernel $Q(f,f)$ is given by the bilinear integral
$Q(f,f) = \int_{\mathbb{R}^2 \times \mathbb{S}^2}B(|v - v_*|) \big[f(v')f(v_*') - f(v) f(v_*)\big]d\omega dv_*$
whose cross section $B$ is given by the Variable Hard Sphere kernel
$ B(|v - v_*|) = C_\alpha |v - v_*|^\alpha, \qquad \alpha \in [0,1]$.
We shall assume that we are working in function spaces which admit a cutoff hypothesis. That is, we have continuity as $\Sigma \rightarrow \infty$ of solutions to the Boltzmann equation with cutoff kernel
$B_\Sigma(|v - v_*|) = \min \big\{ B(|v-v_*|), \Sigma\big\} $.
If you would like to check out how the following simulations are made, check out the Github repository. Also see Lorenzo Pareschi's slides presented at IPAM in 2009.
Simulations
Set $C_\alpha = 1$. These simulations took 2 minutes each, with $N = 50,000$, $\Delta t = 0.01$, $Kn = 1$, and $\alpha = 1$ (hard spheres).
These simulations ran for 5 minutes each with $N = 300,000$, $\Delta t = 0.01$, $Kn = 1$:
This simulation ran for 5 minutes with $N = 300,000$, $\Delta = 0.01$, $Kn = 1$ with initial data consisting of 100 weighted Gaussians with means equidistributed around a circle of radius $= 10$ and standard deviations $= 1$:
I also made a fun extension of these simulations to black and white images. I used the cover art of Daniel Johnston's album Hi, How Are You as an initial condition for the solution. This took 10 minutes with $N = 500,000$, $\Delta t = 0.001$, $Kn =1$, $\alpha = 1$:
