This example presents the propagation of an initially Gaussian wave on a Minkowski space-time with Dirichlet (reflective) boundary conditions. The animations are actually sections of the space with the x = 0 plane, but because the setup is spherically symmetric, they look like waves on a 2D surface. The base grid was made out of 16 × 16 × 16 cells, and the animations were made at level 3. Strictly speaking, level 3 is not fine enough to achieve convergence, but it becomes computationally difficult in terms of run time to go to higher resolution.
This example represents the propagation of an initially Gaussian wave on a compactified Minkowski space-time. Once again the animations are sections of the space with the x = 0 plane, the base grid was made out of 16 × 16 × 16 cells, and the animations were made at level 3. The net effect of the compactification is that the wave becomes cube-like, which is easily noticeable in the animations. Also the wave slows down as it approaches the boundary, and it would take it an infinite time to reach the boundary, since the boundary of the unit cube is the boundary of the Universe in compactified coordinates.
This example shows the x = 0 level 3 section of two initially Gaussian waves on Minkowski space-time, starting from a base grid of 16 × 16 × 16 cells. It can be noticed that although the Gaussians were initially stationary, they tend to "repel". Wall reflection can be observed in the ending frames of the animation, together with a phase jump of &Pi that it incurs.
Some of the material presented on this web page is based upon work supported in part by the Princeton Applied and Computational Mathematics Program and by the National Science Foundation under Grant No. 0745779.