Error estimates for Euler discretization of high-index saddle dynamics

TL;DR

This paper establishes error estimates for Euler discretization of high-index saddle dynamics, addressing challenges from nonlinearity and orthonormalization, and extends results to non-gradient systems.

Contribution

It provides the first rigorous error analysis for Euler discretization of high-index saddle dynamics, including non-gradient systems.

Findings

Error estimates depend on time step size

Method extends to non-gradient saddle dynamics

Supports accuracy of numerical implementations

Abstract

High-index saddle dynamics provides an effective means to compute the any-index saddle points and construct the solution landscape. In this paper we prove error estimates for Euler discretization of high-index saddle dynamics with respect to the time step size, which remains untreated in the literature. We overcome the main difficulties that lie in the strong nonlinearity of the saddle dynamics and the orthonormalization procedure in the numerical scheme that is uncommon in standard discretization of differential equations. The derived methods are further extended to study the generalized high-index saddle dynamics for non-gradient systems and provide theoretical support for the accuracy of numerical implementations.