This is the third installment of the merging black holes posts (if you're just coming in, I suggest you read the intro post). This part will focus on some of the numerical techniques used to simulate black hole mergers. So, if you've ever been interested in how people actually simulate relativity, and what are the current limitations/problems we face, then read on! If not, well, this may be particularly boring for you, ha ha! Just skip the text and look at the pretty pictures (psst... it's what I do most of the time). Just know that when it comes to black holes, especially coalescing binary black holes, numerical relativity is complicated... much like a 12 year old girl's love life.
Bad joke. Anyway, moving on...
Black hole dynamics can be described reasonably well for most of its orbit. For example, analytic postNewtonian expansions can simulate the early inspiral phase, and black hole perturbation methods can treat the late ringdown phase. However, sophisticated computational techniques are required for the late inspiral and merger phases, where, unfortunately (or fortunately?) gravitational waves are at their highest amplitudes.
There are three major hurtles in simulating these highly relativistic systems, which are:
1.
Scale: The gravitational field is incredibly large around the black holes, but gravitational waves themselves are only small perturbations in the gravitational field. Simulations that cover the entire dynamic range from itty bitty waves to huge spacetime warping are extremely hard to write. In addition, simulations cannot deal with singularities themselves, which, unfortunately, is what black holes
are. So it makes it challenging to find a numerical technique that can ignore those regions while still simulating their effects.
2.
Nonlinearity: Einstein’s equations are nonlinear. This means that an element of chaos is introduced into the system. In a chaotic system, vastly different outcomes will occur from similar initial conditions  this is simply what results from nonlinear equations of motion. Therefore, if the initial conditions are not exactly right, the black hole merger may display vastly different characteristics than it would in reality.
3.
Wonky spacetime: current computational models need to decompose Einstein’s equations of General Relativity into equations computers can handle. Einstein’s equations very elegantly combine space and time into one quantity: spacetime. Unfortunately, computers decide to poop out on us. Why? To simulate a system that starts at a specific initial time, computational solutions demand that space and time have to be separated  to be decoupled. There is a way, however, to work around this. Einstein’s equations can be rewritten using the 3+1 decomposition technique. In this technique, the equations are split up into two sets. One set describes the gravitational field at all points within a specific time, and the other set then evolves the gravitational field in time. In this way, once the initial conditions have been set, the code can evolve the merger.
Briefly, there are a couple of different numerical techniques that I know of that work around all these problems: finite difference mesh, spectral or pseudospectral, and adaptive mesh refinement. Finite difference involves equally spaced grids along the system. The partial differential equations are solved in the middle of each grid, and then extrapolated across the boundary. The picture below is an example of this kind of technique.

The grid spacing in this rendition is constant, which is the hallmark of a finite difference simulation. 
The spectral and pseudospectral techniques expand the differential equations in a Fourier series, and the terms that pop out are used as the basis states of the simulation. This simplifies spatial and time derivatives, but also requires the extra step of actually Fourier transforming the field variables.
The last technique (adaptive mesh refinement), uses grids that change size in order to make the simulation more computationally efficient. The simulation essentially determines the number of grid points needed at a specific point in the domain such that the simulation can guarantee accuracy and efficiency. It can also change grid shape or orientation as needed. This can be particularly useful for General Relativity, which is the description of how space becomes warped in the presence of matter (in a gravitational field). This technique is the most commonly used one, although to simulate the full range of the merger, still higher level techniques must be developed.

Yes, the Simpsons got it right! Notice how the grid spacing changes size near the singularity. This is why it's called adaptive mesh refinement. 
It should be noted that adding all spacial
dimensions or throwing away symmetry arguments developed instabilities after
only short integration times!
Next time, I'll talk about what happens when these simulations are implemented. There was a surprise when black holes merged, recoil, which is strictly a consequence of Relativity (and thus reinforces my conviction that Relativity is just plain weird). Stay tuned for more weird black hole fun... I promise to write about it in a timely fashion this time.