Effective Simulation Methods for Structures with Local Nonlinearity: Magnus integrator and Successive Approximations

TL;DR

This paper explores advanced simulation techniques for nonlinear dynamical systems, introducing a novel splitting approach that extends Magnus integrator and compares it with successive approximation methods through stability analysis and numerical experiments.

Contribution

It presents a new splitting method for nonlinear systems, extending Magnus integrator applicability, and compares it with existing successive approximation techniques.

Findings

Magnus integrator can be extended to nonlinear problems with careful exponential computations.

The novel splitting approach improves stability and accuracy in nonlinear simulations.

Numerical experiments demonstrate the effectiveness of the proposed methods.

Abstract

In the following, we discuss nonlinear simulations of nonlinear dynamical systems, which are applied in technical and biological models. We deal with different ideas to overcome the treatment of the nonlinearities and discuss a novel splitting approach. While Magnus expansion has been intensely studied and widely applied for solving explicitly time-dependent problems, it can also be extended to nonlinear problems. By the way it is delicate to extend, while an exponential character have to be computed. Alternative methods, like successive approximation methods, might be an attractive tool, which take into account the temporally in-homogeneous equation (method of Tanabe and Sobolevski). In this work, we consider nonlinear stability analysis with numerical experiments and compare standard integrators to our novel approaches.