Advection-diffusion of a Gaussian Hill

In this example, we will simulate the time evolution of a Gaussian hill in an infinite space with periodic boundaries. The initial form of the Gaussian pulse gives the form: $$C(\mathbf{x}, t=0) = C_0 \exp\left(-\frac{(\mathbf{x} - \mathbf{x}_0)^2}{2\sigma_0^2}\right) + C_{base}$$ The analytical solution with time is: $$C(\mathbf{x}, t) = \frac{\sigma_0^2}{\sigma_0^2 + \sigma_D^2} C_0 \exp \left( -\frac{(\mathbf{x} - \mathbf{x}_0)^2}{2(\sigma_0^2 + \sigma_D^2)} \right) + C_{base}$$

In our simulation, we define $nelem = 500$ which generates a 1000*1000 2D mesh. The mesh is recognized
as an infinite area for $\sigma_0=40$. The D2Q9 layout is used. Boundaries from all directions are periodic. As the area is much larger than the $ \sigma_0 $, the effect from the boundaries is neglected. Results of t=200 are used to calculate the diffusion factor.

In ./musubi.lua, define 'shape' in 'tracking' block as

shape = {
  -- kind = 'all'
  kind  = 'canoND',
  object  = {
    origin = {-nelem-1, 0, 0},
    vec = { {2.0*(nelem+1), 0., 0.0} }
  }
},

shape = {
  kind = 'all'
  -- kind  = 'canoND',
  -- object  = {
  --   origin = {-nelem-1, 0, 0},
  --   vec = { {2.0*(nelem+1), 0., 0.0} }
  -- }
},