A finite-difference method is developed for solving two coupled, ordinary differential equations that model a sequence of chemical reactions. The initial-value problem is highly nonlinear and involves three parameters. Various types of theoretical solution of this problem (the Sal'nikov thermokinetic oscillator problem) may be found, depending on these parameters; this is because the stationary point is surrounded by up to two limit cycles. The well-known, first-order, explicit Euler method and an implicit finite-difference method of the same order are used to compute the solution. It is shown that this implicit method may, in fact, be used explicitly and extensive numerical experiments are made to confirm the superior stability properties of the alternative method.