A Class of Power Series Collocation Multistep Methods with Legendre Interpolant for the Integration of Initial Value Problems

Abstract

In this paper, we present a new approach for the derivation of a class of collocation linear multistep methods for the integration of some initial value problems of ordinary differential equations. The derivation of the discrete methods of orders p=2, 3, 4, and 5 for the un-economized collocation method for solving Initial Value Problems. By introducing the Legendre polynomial terms, the economized invariants of the methods are derived. The methods are all consistent and zero stable, thus they are convergent. For implementation as predictor-corrector pairs, the class of explicit methods of the Adams Bashforth methods are used as predictors while the new methods, which are implicit, are used as correctors. Python programming language and MATLab were used for the numerical computations. The methods exhibits the A-stability properties of the standard Trapezoidal method and performs better when implemented to solve Stiff initial value problems.

Keywords: Linear Multistep Methods, Legendre polynomials, integration, Collocation methods and Absolute stability