This paper describes the development and application of the fully implicit finite-difference algorithm with the Richtmyer modification (the FIRM algorithm) and an exponential time grid to simulations of stiff problems involving homogeneous kinetics and simulations of stiff problems involving semi-infinite linear diffusion coupled with heterogeneous and homogeneous kinetics. The accuracy and stability of the FIRM algorithm with various levels and the dependence upon the starting protocol are discussed in detail.