Linearization methods in classical and quantum mechanics

The applicability and accuracy of linearization methods for initial-value problems in ordinary differential equations are verified on examples that include the nonlinear Duffing equation, the Lane–Emden equation, and scattering length calculations. Linearization methods provide piecewise linear ordinary differential equations which can be easily integrated, and provide accurate answers even for hypersingular potentials, for which perturbation methods diverge. It is shown that the accuracy of linearization methods can be substantially improved by employing variable steps which adjust themselves to the solution.