Project 9

Wave Equation in one dimension

On Eddington-Finkelstein Background

Implicit Method (9 point)

with Radiation Boundary Conditions

General Results

Simiulation uses in-going Eddington-Finkelstein coordinates.
Simulation successfully tracks an in going pulse across the event horizon.
A wave that begins out side the horizon slowly gets away, and broadens out significantly.

Follow the excitement of an ingoing pulse. This pulse begins near the event horizon, and travels radially inward. The event horizon is at r = 1

Initial data
Wave approaches event horizon
Wave passing through event horizon Not the best decision!
Wave completely inside the event horizon IT'S TOO LATE!
Wave leaves grid, searching out new and unexplored territory near the singularity. Alas, you, poor viewer will never see it . . . (Infinite coordinate time and all that.)

Follow the excitment of an outgoing pulse. This pulse begins near the event horizon, but barely makes it away (close call!). Again, the event horizon is at r=1.

The wave moves very slowly, and this simulation requires many more time steps (1000 iterations here). The wave spreads out in time.

Family Portrait. Here are several snapshots plotted together. The left most wave is the initial data. Each wave to the right is a snapshot from later times. Notice how the waves broaden while moving outward
Wave passes boundary. This plot shows the wave at three instances while passing through the right grid boundary. (Not all that exciting I'll grant, but just to show that it can be done.)

Here is an outward moving wave centered on the event horizon. The plot shows a superposition of the the wave a four different times. The central wave is the initial data, and as time increases the wave spreads in both directions, albeit not symmetrically.

Still under construction!!

To be added next: ??