Many scientific and engineering problems need to use numerical methods and algorithms to obtain computational simulation results because analytical solutions are seldom available for them. The chemical dissolution-front instability problem in fluid-saturated porous rocks is no exception. Since this kind of instability problem has both the conventional (i.e. trivial) and the unconventional (i.e. nontrivial) solutions, it is necessary to examine the effects of different numerical algorithms, which are used to solve chemical dissolution-front instability problems in fluid-saturated porous rocks. Toward this goal, two different numerical algorithms associated with the commonly-used finite element method are considered in this paper. In the first numerical algorithm, the porosity, pore-fluid pressure and acid/solute concentration are selected as basic variables, while in the second numerical algorithm, the porosity, velocity of pore-fluid flow and acid/solute concentration are selected as basic variables. The particular attention is paid to the effects of these two numerical algorithms on the computational simulation results of unstable chemical dissolution-front propagation in fluid-saturated porous rocks. The related computational simulation results have demonstrated that: 1) the first numerical algorithm associated with the porosity-pressure-concentration approach can realistically simulate the evolution processes of unstable chemical dissolution-front propagation in chemical dissolution systems. 2) The second numerical algorithm associated with the porosity-velocity-concentration approach fails to simulate the evolution processes of unstable chemical dissolution-front propagation. 3) The extra differential operation is the main source to result in the failure of the second numerical algorithm.