An efficient step size selection for ODE codes

TL;DR

The paper addresses efficient step-size control for non-stiff IVPs by introducing a (p,p-1) method-based rule in which the next step size is $h_{n+1}=h_n\left(\tau/(|\epsilon|h_n)\right)^{1/(p+1)}$, complemented by bounds on $h_{n+1}$ and an acceptance criterion. This scheme is designed to be asymptotically independent of the current step size as $h_n\to0$ and is compared against the standard $h_{n+1}=h_n\left(\tau/|\epsilon|\right)^{1/(p+1)}$ rule. The authors optimize parameters $(\sigma,\lambda_1,\lambda_2)$ using the DETEST benchmark with an embedded RK $(5,4)$ and report reductions in function evaluations in a substantial fraction of problems, demonstrating practical efficiency gains for non-stiff ODE solvers. The work provides concrete parameter guidelines and validates improvements across multiple test cases, highlighting the method's potential to enhance solver performance in real-world computations.

Abstract

We give an algorithm for efficient step size control in numerical integration of non-stiff initial value problems, based on a formula tailormade to methods where the numerical solution is compared with a solution of lower order.