Adaptive Numerical Quenching for a Nonlocal Neumann Problem with Nonhomogeneous Flux

Abstract

We analyze an adaptive approximation of quenching for a nonlocal parabolic problem with a nonhomogeneous Neumann-type flux. The flux is treated as a nonnegative exterior input, while the singular source is controlled by a mesh-scaled barrier time step. The discrete scheme preserves the admissible range, quenches in finite accumulated time, and satisfies upper and lower estimates for the numerical quenching time. We also prove convergence before quenching and convergence of fixed-level detection times. The numerical tests verify the invariant bound, the monotone influence of the flux intensity, and the second-order spatial behavior of the detected time.