Note: This section uses the
Einstein summation convention, and the
Misner-Thorne-Wheeler units and metric conventions: the speed of light is assumed unity,
and the metric signature is -+++. The determinant of the metric is denoted by g, and partial derivatives
are denoted by commas.
Consider the wave equation on an arbitrary space-time equipped with metric gab and with an arbitrary source term F:
This differential equation is second order in time, and to solve it numerically
two techniques can generally be used. Either the equation is solved "as is," using a three
time level discretization scheme, or it is split into two equations that are first order in
time, which are then solved with a two time level discretization scheme. The latter approach,
when applicable, generally works better, so it will be presented here.
The trick to splitting the wave equation into two first order in time equations is to define the "time component of the momentum four-vector" as:
This expression can be inverted to obtain:
Using the definition of and the above expression, the wave equation can be expanded as:
Thus, the wave equation has been reduced to two differential equations that are first order in time.
Solving PDEs requires imposing boundary conditions on the edges of the region on which the PDEs are being solved. These boundary conditions can sometimes
lead to pathological behavior, as they may cause unphysical effects. For instance, Dirichlet boundary conditions usually reflect high
frequency modes, which can introduce high frequency oscillations of large amplitude to appear in the PDE solution. An elegant fix to the boundary condition
problem is coordinate compactification, which essentially allows solving the PDEs on the entire Universe.
Coordinate compactification works in the following way: Suppose our coordinates are t, x, y, z, each ranging from -&infin to +&infin. Then the transformation to coordinate system t', x', y', z' given by:
is called a coordinate compactification. This transformation essentially compactifies the entire Universe to the unit cube. Dirichlet boundary conditions
can now be imposed without any problems, as the wave will never reach the boundary of the cube, so high frequency resonance will no longer happen.
Under the coordinate transformation, the metric transforms as:
For Minkowski space-time, this gives:
The square root of the determinant of the compactified Minkowski metric is:
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.