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  and , which we refer to, along with , 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.
Living Rev. Relativity 15, (2012), 9
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