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 [228Jump To The Next Citation Point]. [227Jump To The Next Citation Point] 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) [229Jump To The Next Citation Point] 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.

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