The Stochastic TR-BDF2 Scheme of Order 2
Abstract
Our main objective in this paper is to develop a second-order stochastic numerical method which generalizes the well-known deterministic TR-BDF2 scheme. Since most stochastic techniques used for approximating the solution of a stochastic differential equation may have lower order compared to the deterministic case, we have elaborated a scheme which not only preserves the second-order accuracy of the original scheme in the stochastic framework, but also its -stability. Once we obtain the scheme and prove its second-order accuracy and -stability, which is not a trivial task, we also state a result concerning its -stability. This concept is also analyzed for different parameter ranges in our scheme and the It{\^o}--Taylor approximation of order 2, revealing scenarios where, for certain time step sizes, the developed method is -stable while the It{\^o}--Taylor one is not. This concept is really useful to tackle slow-fast problems such as stiff ones, which we aim to explore further in future work. Finally, we validate the theoretical results with some academic test cases.
Cite
@article{arxiv.2602.10773,
title = {The Stochastic TR-BDF2 Scheme of Order 2},
author = {Tomás Caraballo and Macarena Gómez-Mármol and Ignacio Roldán},
journal= {arXiv preprint arXiv:2602.10773},
year = {2026}
}