Numerical self-force calculations for scalar particles, formulated in the lab frame

We derive equations of motion for scalar particles self-consistently interacting with a scalar field,including the radiation produced by the particles' acceleration. Our approach differs in three key aspects from current methods: (1) we assume a small but finite discretization length scale $h$, which allows us to treat the particle as a small but finite object, (2) we choose the state vector for the system before deriving equations of motion, and (3) we formulate the equations explicitly in the lab frame and not in a manifestly covariant manner. This approach, which is self-consistent, happens to greatly simplify the resulting equations and their derivation, and is directly suitable for numerical calculations. The result is an effective source method which generalizes to electrodynamics or general relativity in a straightforward manner (although we do not consider this here). We then provide two possible discretizations of these equations, based on finite volumes and spectral methods, and show results of one-dimensional calculations. These calculations show excellent agreement with analytic solutions.