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We present a new method designed to avoid numerical challenges that haveimpeded calculation of the Lorenz gauge self-force acting on a compact objectinspiraling into a Kerr black hole. This type of calculation is valuable increating waveform templates for extreme mass-ratio inspirals, which are animportant source of gravitational waves for the upcoming Laser InterferometerSpace Antenna mission. Prior hyperbolic partial differential equation (PDE)formulations encountered numerical instabilities involving unchecked growth intime; our new method is based on elliptic PDEs, which do not exhibitinstabilities of that kind. For proof of concept, we calculate the self-forceacting on a scalar charge in a circular orbit around a Kerr black hole. Weanticipate this method will subsequently facilitate calculation of first-orderLorenz gauge Kerr metric perturbations and self-force, which could serve as afoundation for second-order Kerr self-force investigations.

New self-force method via elliptic partial differential equations for Kerr inspiral models