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A high order compact scheme is employed to obtain the numerical solution of a singular, one-dimensional, reaction-diffusion equation of the quenching-type motivated by models describing combustion processes. The adaptation of the temporal step is discussed in light of the proposed theory. A condition, reminiscent of the Courant-Friedrichs-Lewy (CFL) condition, is determined to guarantee that the numerical solution monotonically increases, a property the analytic solution is known to exhibit. Strong stability is proven in a Von-Neumann sense via the 2-norm. Computational examples illustrate the spatial convergence and quenching times are calculated for particular singular source terms.