In the last years, due to the progress of the art and the increasing potentiality for the efficient calculation of the matrix exponential, the method classes of exponential and Magnus integrators received a lot of attention. In particular, owing to their favourable stability and convergence properties, numerical discretisations based on the Magnus expansion are an approved method class for the time integration of Schrödinger equations with time-dependent Hamiltonian.   

In this talk, I will consider Magnus type integrators for the time discretisation of linear and nonlinear parabolic initial-boundary value problems and study their stability and convergence behaviour. For the analysis, it is convenient to interprete the partial differential equation as an abstract evolution equation on a function space employing the framework of sectorial operators and analytic semigroups on Banach spaces. The theoretical results are illustrated and confirmed by  various applications and numerical examples.

We present efficient high order numerical schemes for the strong solution of linear stochastic differential equations. These schemes are based on the Neumann and Magnus expansions and the efficient evaluation of high order multiple Stratonovich integrals. Comparing these methods to a classical stochastic numerical integration scheme, we demonstrate several orders of magnitude improvement in accuracy for the same computational cost. To indicate their preferential use in some dynamic programming applications, we apply them to a stochastic Riccati differential system that can be linearized.

We briefly review the classical stability theory for Runge&endash;Kutta methods and comment on its applicability to the Magnus method. Then, we consider the behaviour of the fourth-order Magnus method on a particular linear and mildly nonautonomous equation. The matrix in the equation has one eigenvalue around zero and one very negative eigenvalue. We calculate estimates for the local and global error and conclude that order reduction takes place. The analysis is supported by numerical experiments.

We explore the numerical properties of the the Exponential fourth order Lawson integrator on the non-linear Schr\"odinger equation for varying regularity of the potential and the initial condition. Some estimates on the regularity dependency are presented.

It has been know since the time of Jacobi that the solution to the free rigid body (FRB) equations of motions is given in terms of a certain type of elliptic functions. It is an interesting question to ask whether these functions can be calculated in such a way that they yield a faster and more accurate solutions to the FRB equations than standard numerical ODE solvers.

In this talk we give a brief review of the derivation of the solution to the FRB equations in terms of the Jacobi elliptic functions. In particular we will recount how these function can be efficiently approximated, and show some numerical tests which compares the performance with the method of McLachlan and Reich. Finally, we will dicuss possible extensions and limitations of this approach.