On the numerical solution of the Van der Pol equation using collocation

TL;DR

This paper explores the use of collocation methods for solving the Van der Pol equation, identifies limitations in the original approach due to computational costs, and proposes a segmented collocation adaptation that improves efficiency.

Contribution

The paper introduces a segmented collocation method tailored for the Van der Pol equation, enhancing computational efficiency over traditional collocation techniques.

Findings

Original collocation method is computationally expensive for stiff equations.

Segmented collocation achieves similar accuracy with lower costs.

Theoretical and numerical evidence supports the effectiveness of the segmented approach.

Abstract

This study introduces the reader to the theory of approximating the solution(s) of a non-linear, second order, ordinary differential equation (ODE) with piecewise polynomial functions by using the collocation method. It then focuses on the application of the method to generate collocation approximations for the Van der Pol equation with initial conditions and with varying equation parameter, {\mu}. It is shown that the collocation method in its original form is impractical for generating these approximations due to the numerical costs of producing such an approximation, particularly as the stiffness of the equation increases with parameter, {\mu}. An adaptation to the method, termed segmented collocation, is proposed, reliant upon on the specific structure and convergence for initial value problems, and shown theoretically and numerically to be capable of generating equivalent approximations for much superior costs. Further study is proposed; to compare the segmented method to other numerical methods traditionally employed with equations of this type such as the Runge-Kutta family; and to develop and test the method in order to ascertain its complete usefulness and competitive standing in the context of approximating second order, ODEs with initial conditions.