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Dumbbell shaped region (starting from the axis)


This region has two connected cavities, with a ball inside larger cavity. Wave function spreads from the axis to the entire region.

Equation to solve

i ∂Ψ(x,y,z)

+ ∆Ψ(x,y,z) + |Ψ(x,y,z)|2Ψ(x,y,z) − 0.3 |Ψ(x,y,z)|4Ψ(x,y,z) = 0


The region is bounded by a rectangular box [−1,1]×[−1.5,3]×[−1,1]. Large cavity is a ball of radius 1 with the center at (0,2,0). Small cavity is a ball of radius 0.5 with the center at (0,−1,0). Connector is a cylinder of radius 0.2 with centers of its bases in the centers of the balls. A small ball inside larger cavity is centered at (0,2,0.5), and has radius 0.2.


Mesh grading of the rectangular box is 20×45×20 was obtained with use of blockMesh utility. Mesh of the dumbbell shaped region was obtained by SnappyHexMesh utility.

Initial condition

The function at zero time is a rotationally symmetric Gaussian distribution centered at symmetry axis,
Ψ(x,y,z) = e−9x2−9z2.


Boundary conditions

Neumann (zero gradient normal to the boundary).


The time step is ∆t = 10−4.



Spreading of wave function from the axis. The dumbbell is sliced to show the interior.


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