High order second derivative multistep collocation methods for ordinary differential equations

Document Type : Research Article

Author

• Somayyeh Fazeli

Marand Technical Faculty, University of Tabriz, Tabriz, Iran.

10.22067/ijnao.2024.85789.1358

Abstract

In this paper, we introduce second derivative multistep collocation methods for the numerical integration of ordinary differential equations. These methods combine the concepts of both multistep and collocation methods, using the second derivative of solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties i.e. $A$-stability for  ODEs. Using the second-order derivatives leads to a high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the $r$ previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the collocation methods. We present the construction of the technique, discuss the analysis of the order of accuracy and linear stability properties, and provide some numerical results confirming the theoretical expectations. A stiff system of ordinary differential equations, the Robertson chemical kinetics problem, and the two-body Pleiades problem are considered case studies for comparing the achievable accuracy of the proposed methods with existing methods.

Keywords

• Collocation
• Linear stability
• Ordinary differential equation
• Second derivative methods

Main Subjects

• Numerical methods for differential equations and its applications

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