Linearized Θ-methods I. Ordinary differential equations Original Research Article

Fully-linearized Θ-methods for autonomous and non-autonomous, ordinary differential equations are derived by approximating the non-linear terms by means of the first-degree polynomials which result from Taylorʹs series expansions. These methods are implicit but result in explicit solutions. A-stable, consistent and convergent: however, they may be very demanding in terms of both computer time and storage because the matrix to be inverted is, in general, dense. The accuracy of fully-linearized Θ-methods is comparable to that of the standard, implicit, iterative Θ-methods, and deteriorates as the value of Θ is decreased from Θ = 0.5, for which both Θ- and fully-linearized Θ-methods are second-order accurate. Partially-linearized Θ-methods based on the partial linearization of non-linear terms have also been developed. These methods result in diagonal or triangular matrices which may be easily solved by substitution. Their accuracy, however, is lower than that of fully-linearized Θ-methods.