Local Stability of Euler's and Kahan's Methods

Abstract

Two discretization methods, the forward Euler's method and the Kahan's reflexive method, are compared by looking at the local stabilities of fixed points of a system of differential equations. We explain why forward Euler's method is not as good from the viewpoint of complex analysis. Conformal mappings are used to relate the eigenvalues of the Jacobian matrices of the differential equations system and the resulting difference equations system. The Euler's method will not preserve Hopf bifurcation. The Kahan's method preserves the local stability of the fixed points of the differential equations.