Karolina Kropielnicka

Arieh Iserles, Karolina Kropielnicka, Pranav Singh

The computation of the Schrödinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.

Arieh Iserles, Karolina Kropielnicka, Pranav Singh

Numerical solutions for laser-matter interaction in Schrödinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small cost, to extend any fourth-order scheme for Schrödinger equation with time-indepedent potential to a fourth-order method for Schrödinger equation with laser potential. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.

Arieh Iserles, Karolina Kropielnicka, Pranav Singh

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of high-order exponential splitting schemes that are able to overcome these challenges by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. This allows us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions.

José A. Carrillo, Piotr Gwiazda, Karolina Kropielnicka et al.

The Escalator Boxcar Train (EBT) method is a well known and widely used numerical method for one-dimensional structured population models of McKendrick-von Foerster type. Recently the method, in its full generality, has been applied to aged-structured two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. We derive the simplified EBT method and prove its convergence to the solution of Fredrickson-Hoppensteadt model. The convergence can be proven, however only if we analyse the whole problem in the space of nonnegative Radon measures equipped with bounded Lipschitz distance (flat metric). Numerical simulations are presented to illustrate the results.

Philipp Bader, Arieh Iserles, Karolina Kropielnicka et al.

We build an efficient and unitary (hence stable) method for the solution of the semi-classical Schrödinger equation subject with explicitly time-dependent potentials. The method is based on a combination of the Zassenhaus decomposition (Bader, Iserles, Kropielnicka & Singh 2014) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.