### 7.6 Remarks

The classical reference for the stability theory of finite-difference for time-dependent problems is [361].
A modern account of stability theory for initial-boundary value discretizations is [228]. [227] includes a
discussion of some of the main stability definitions and results, with emphasis on multiple aspects of
high-order methods, and [415, 416] many examples at a more introductory level. We have omitted
discussing the discrete version of the Laplace theory for IBVP, developed by Gustafsson, Kreiss
and Sundström (known as GKS theory or GKS stability) [229] since it has been used very
little (if at all) in numerical relativity, where most stability analyses instead rely on the energy
method.
The simplest stability analysis is that of a periodic, constant-coefficient test problem. An eigenvalue
analysis can include boundary conditions and is typically used as a rule of thumb for CFL limits or to
remove some instabilities. The eigenvalues are usually numerically computed for a number of different
resolutions. See [171, 175] for some examples within numerical relativity.

Our discussion of Runge–Kutta methods follows [96] and [230], which we refer to, along
with [231], for the rich area of methods for solving ordinary differential equations, in particular
Runge–Kutta ones. We have only mentioned (one-step) explicit methods, which are the ones
used the most in numerical relativity, but they are certainly not the only ones. For example,
stiff problems in general require implicit integrations. [274, 322, 273] explored implicit-explicit
(IMEX) time integration schemes in numerical relativity. Among many of the topics that we have
not included is that of dense output. This refers to methods, which allow the evaluation of
an approximation to the numerical solution at any time between two consecutive timesteps,
at an order comparable or close to that of the integration scheme, and at low computational
cost.