A second-order explicit method is developed for the numerical solution of the initialvalue problem w'(r) = d w ( t ) / d t = + ( w ) , t > 0, w ( 0 ) = Wo, in which the function I $ ( w ) = a w ( 1 -w ) ( wa ) , with a and a real parameters, is the reaction term in a mathematical model of the conduction of electrical impulses along a nerve axon. The method is based on four first-order methods that appeared in an earlier paper by Twizell, Wang, and Price [Proc. R. SOC. (London) A 430, 541-576 (1990)]. In addition to being chaos free and of higher order, the mcthod is seen to converge to one of the correct steady-state solutions at w = 0 or w = 1 for any positive value of a . Convergence is monotonic or oscillatory depending on W o , a , a, and I, the parameter in the discretization of the independent variable t . The approach adopted is extended to obtain a numerical method that is second order in both space and time for solving the initial-value boundary-value problem au/ar = K a 2 u / a x 2 + a u ( 1 -u ) ( ua ) in which u = u ( x , t ) . The numerical method so developed obtains the solution by solving a single linear algebraic system at each time step.