The Stochastic TR-BDF2 Scheme of Order 2

TL;DR

This paper introduces a second-order stochastic numerical scheme, extending the deterministic TR-BDF2 method, which maintains high accuracy and stability properties for solving stochastic differential equations, especially stiff problems.

Contribution

The paper develops and proves a second-order, $A$- and $MS$-stable stochastic extension of the TR-BDF2 scheme, a novel advancement in stochastic numerical methods.

Findings

The scheme achieves second-order accuracy in stochastic settings.

It maintains $A$-stability and $MS$-stability under certain conditions.

The method outperforms Itô--Taylor in stability for specific step sizes.

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 $A$-stability. Once we obtain the scheme and prove its second-order accuracy and $A$-stability, which is not a trivial task, we also state a result concerning its $MS$-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 $MS$-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.