Curing the self-force runaway problem in finite-difference integration

The electromagnetic self-force equation of motion is known to be afflicted by the so-called runaway problem. A similar problem arises in the semiclassical Einstein field equation and plagues the self-consistent semiclassical evolution of spacetime. Motivated to overcome the latter challenge, we first address the former (which is conceptually simpler) and present a pragmatic finite-difference method designed to numerically integrate the self-force equation of motion while curing the runaway problem. We first restrict our attention to a charged point-like mass in a one-dimensional motion, under a prescribed time-dependent external force. We demonstrate the implementation of our method using two different examples of an external force, for which our numerical results agree with those obtained by two other methods (a Dirac-type solution and a reduction-of-order solution). Next, we extend our treatment to general higher-order linear ODEs with a radiation-reaction term. These equations might be non-homogeneous and generally include the case of an external force which depends also on the position. We then close by further extending our analysis to a class of suitably well-behaved nonlinear equations. Both the linear and nonlinear cases are demonstrated in simple physical systems and match the solutions obtained by another method (reduction-of-order). All cases demonstrate a complete suppression of the undesired runaway mode, along with an accurate account of the radiation-reaction effect at the physically relevant time scale- thereby illustrating the effectiveness of our method in curing the self-force runaway problem.