Good source for numerical simulations of Wigner function?
Your equation in the Liouville form is elementary for numerical integration, it is structurally just a linear advection equation with spatially varying coefficients. The transformed equation with the kernel F is not useful at all for numerical solution, don't bother with it.
All we have here is a 2D advection equation (I use y instead of p):
$$ \partial_{t} W(x,y,t) = V_x(x,y) \partial_{x} W(x,p,t) + V_y(x,y) \partial_{x} W(x,p,t) $$
where $V_x=C_1 y$ and $V_y=C_2 x$.
A simple way to solve it is using explicit time-stepping for the spatially and temporally discretized form
$$ (W_{i,j}^{k+1}-W_{i,j}^{k})/\tau = V_x (W_{i,j}^{k}-W_{i-1,j}^{k})/\delta x + V_x (W_{i,j}^{k}-W_{i,j-1}^{k})/\delta y $$
This elementary time-stepping scheme will allow time integration; as long as you provide boundary conditions upstream you can march along the characteristics. Note that in this method some constraints on the time step will arise from the stability requirements, and the accuracy order is low. Also note that the upwind spatial discretization is used for the advection operators, otherwise it will be unstable for any time step. You can improve this a bit by using explicit Lax-Wendroff or some other classical scheme for hyperbolic PDEs. However, with today's computers, you can as easily use implicit time-stepping and just solve for the whole domain as a linear system of equations. To improve the accuracy I would go to the 4th order in the spatial discretization. Note that $(x=0,y=0)$ is a singular point here so some care should be taken to avoid using it in the numerical stencil.
Well, for the harmonic oscillator, you have the full closed answer, so you don't really need numerics. It is, as Groenewold discovered in his 1946 thesis (Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.), merely rigid rotation!:
$$W(x,p; t) = W(x \cos t − p \sin t, p \cos t + x \sin t; 0).$$
Have set $m=ω=1$ for simplicity.
A state of the art numerical simulation example this year is
- Shao & Sellier, "Comparison of deterministic and stochastic methods for time-dependent Wigner simulations", J. Comput. Phys. 300 167-185 (2015).
The following classic references rely on numerical simulation to draw sound conclusions with useful insights into WF time evolution, and include links to numerical sources:
O. Steuernagel, D. Kakofengitis, and G. Ritter, "Wigner Flow Reveals Topological Order in Quantum Phase Space Dynamics", Phys. Rev. Lett. 110, 030401 (2013), arXiv:1208.2970.
O. Gat, "Quantum dynamics and breakdown of classical realism in nonlinear oscillators", J. Phys. A: Math. Theor. 40 F911 (2007), arXiv:quant-ph/0702191.
T. Dittrich, E. A. Gomez, and L. A. Pachon, "Semiclassical propagation of Wigner functions", J. Chem. Phys. 132, 214102 (2010), arXiv:0911.3871.
T. Dittrich, C. Viviescas, and L. Sandoval, "Semiclassical Propagator of the Wigner Function", Phys. Rev. Lett. 96, 070403 (2006), arXiv:quant-ph/0508057.