A point-particle-limit effective-source approach for computing gravitational self-force in the Lorenz gauge

The traditional effective-source method is hampered by complex analytical expressions and the inherent smoothness limitation, which result in high computational costs and complicate its implementation. To overcome these limitations, we introduce the point-particle-limit effective source method, which analytically takes the size of the effective source to zero, thereby transforming the problem into a well-defined jump condition for the retarded metric field at the particle position, governed by the local singular field. This formulation naturally pairs with a discontinuous Galerkin scheme, whose inherent capability to accommodate solution discontinuities enables highly accurate enforcement of the jump conditions. We apply both the traditional and point-particle-limit effective source methods to calculate the time-domain gravitational metric perturbation and gravitational self-force in the Lorenz gauge for a point particle in a circular orbit around a Schwarzschild black hole. The point-particle-limit effective source method achieves relative errors below 10^-4 (temporal) and 10^-3 (radial) in the gravitational self-force at \ell_\rm max=15 for circular Schwarzschild orbits, reduces the computational cost by an order of magnitude compared with the traditional approach, and yields energy fluxes that agree with frequency-domain benchmarks. The numerical results demonstrate the advantages of the point-particle-limit effective source method, validating its effectiveness and efficiency and thereby establishing a numerical foundation for computing generic geodesic orbits or long-time self-consistent orbital evolution.