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

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.