An improved theta-method for systems of ordinary differential equations

The θ-method of order 1 or 2 (if θ=1/2) is often used for the numerical solution of systems of ordinary differential equations. In the particular case of linear constant coefficient stiff systems the constraint 1/2 ≤ θ ≤1, which excludes the explicit forward Euler method, is essential for the method to be A -stable. Moreover, unless θ=1/2, this method is not elementary stable in the sense that its fixed-points do not display the linear stability properties of the fixed-points of the involved differential equation. We design a non-standard version of the θ-method of the same order. We prove a result on the elementary stability of the new method, irrespective of the value of the parameter θ ∈[0,1]. Some absolute elementary stability properties pertinent to stiffness are discussed.