Crank-Nicolson method for the numerical solution of models of excitability

We analyze a Crank-Nicolson scheme for a family of nonlinear parabolic partial differential equations. These equations cover a wide class of models of excitability, in particular the Hodgkin-Huxley equations. To do the analysis, we have in mind the general discretization framework introduced by Lopez-Marcos and Sanz-Serna [in Numerical Treatment of Differential Equations, K. Strehemel, Ed., Teubner-Texte zur Mathematik, Leipzig, 1988, p. 216]. We study consistency, stability and convergence properties of the scheme. We use a technique of modified functions, introduced by Strang [Numer. Math. 6, 37 (1964)], in the study of consistency. Stability is derived by means of the energy method. Finally we obtained existence and convergence of numerical approximations by means of a result due to Stetter (Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin, 1973). We show that the method has optimal order of accuracy in the discrete H1 norm.