TITLE:
A Filtered Milne-Simpson ODE Method with Enhanced Stability Properties
AUTHORS:
Ari Aluthge, Scott A. Sarra
KEYWORDS:
Ordinary Differential Equations, Numerical Methods, Weak Instability
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.11 No.1,
January
31,
2023
ABSTRACT: The Milne-Simpson method is a two-step implicit linear multistep method for the numerical solution of ODEs that obtains the theoretically highest order of convergence for such a method. The stability region of the method is only an interval on the imaginary axis and the method is classified as weakly stable which causes non-physical oscillations to appear in numerical solutions. For this reason, the method is seldom used in applications. This work examines filtering techniques that improve the stability properties of the Milne-Simpson method while retaining its fourth-order convergence rate. The resulting filtered Milne-Simpson method is attractive as a method of lines integrator of linear time-dependent partial differential equations.