Abstract
M.Sc. (Applied Mathematics)
Explicit schemes for integrating ODEs and time–dependent partial differential
equations (in the method of lines–MoL–approach) are very well–known
to be stable as long as the maximum sizes of their timesteps remain below
a certain minimum value of the spatial grid spacing. This is the Courant–
Friedrich’s–Lewy (CFL) condition. These schemes are the ones traditionally
being used for performing simulations in Numerical Relativity (NR). However,
due to the above restriction on the timestep, these schemes tend to be so
much inadequate for simulating some of the highly probable and astrophysically
interesting phenomenae. So, it is of interest this currernt moment to
seek or find integrating schemes that may help numerical relativists to somehow
circumvent the CFL restriction inherent in the use of explicit schemes.
In this quest, a more natural starting point appears to be implicit schemes.
These schemes possess a highly desireable stability property – they are unconditionally
stable. There also exists a combination of implicit and explicit
(IMEX) schemes. Some researchers have already started exploring (since
2009, 2011) these for NR purposes.
We report on the implementation of two implicit schemes (implicit Euler,
and implicit midpoint rule) for Einstein’s evolution equations. For low computational
costs, we concentrated on spherical symmetry. The integration
schemes were successfully implemented and showed satisfactory second order
convergence patterns on the systems considered. In particular, the Implicit
Midpoint Rule proved to be a little superior to the implicit Euler scheme.