  In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been
introduced to solve time-implicit schemes of reaction-diffusion equations. In
this research, we conduct theoretical studies on this first-order algorithm
equipped with a quadratic regularization term. We provide sufficient conditions
under which the proposed algorithm and its time-continuous limit converge
exponentially fast to a desired time-implicit numerical solution. We show both
theoretically and numerically that the convergence rate is independent of the
grid size, which makes our method suitable for large-scale problems. The
efficiency of our algorithm has been verified via a series of numerical
examples conducted on various types of reaction-diffusion equations. The choice
of optimal hyperparameters as well as comparisons with some classical
root-finding algorithms are also discussed in the numerical section.
