Multi-Order Runge-Kutta Methods or how to numerically solve initial value problems of any order
Year: 2025
Type: preprint
Abstract: When one wishes to numerically solve an initial value problem, it is customary to rewrite it as an equivalent first-order system to which a method, usually from the class of Runge-Kutta methods, is applied. Directly treating higher-order initial value problems without such rewriting, however, allows for significantly greater accuracy. We therefore introduce a new generalization of Runge-Kutta methods, multi-order Runge-Kutta methods, designed to solve initial value problems of arbitrary order. We establish fundamental properties of these methods, including convergence, order of consistency, and linear stability. We also analyze the structure of the system satisfied by the approximations of a method, which enables us to provide a proper definition of explicit methods and gain a finer understanding of implicit methods.
Source: arXiv (Cornell University)
Author: Petronijevic, Loris
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415332483
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