# Adaptive integration of evolution equations

W. Auzinger, Institute for Analysis and Scientific Computing

Many numerical techniques for the integration of time-dependent problems are based on constant stepsize sequences. On the other hand, most methods of one-step type allow for an easy change of stepsize. This is only reasonable if it is controlled according to a given accuracy requirement, assuming that a reliable, computable measure for the accuracy of a single step (typically an estimate for the local error of such a step) is available. Such algorithms and the underlying theory were introduced in the context of splitting methods applied to linear and nonlinear Schrödinger equations in [1-4], where also numerical results are reported. Other ways of estimating local error are also the topic of current investigations. Naturally, also adaptivity in space is a relevant topic. In the context of a stationary singular ODE boundary value problem, such a technique was introduced and analyzed in [5].

# PhD Project 1: Adaptive exponential Krylov propagators

Co-supervisor: Peter Blaha

At the core of most algorithms for numerical time propagation it is necessary to approximate the action of an exponential map applied to a given state. Mathematically speaking this means evaluation of the exponential of a large (usually sparse) matrix applied to a given vector. Except in special cases this is usually accomplished using Krylov subspace methods. For controlling the accuracy of such an approximation, different techniques have been proposed. These are mainly heuristic or apply only to special cases (like unitary propagation).

In this project the topic is to be investigated further, in the relevant application context. In particular, the idea of using the defect of a Krylov approximation has been proposed before, but it can be extended further in order to obtain a more reliable error indicator. Taking account of the internal properties of the approximation, improved multi-defect-based error indicators will be designed, analyzed, implemented, and tested.

# PhD Project 2: Adaptive propagators for time-dependent evolution systems

Co-supervisor: Florian Libisch

For a large evolution system with coefficients depending on time, numerical propagation is much more challenging that for the case of constant coefficients. For linear problems, for instance Schrödinger equations with time-dependent potentials, different techniques have been proposed, e.g., commutator-free Magnus integrators. However, the adaptive selection of time steps for a given tolerance has not been systematically studied. This topic is to be investigated, also in a relevant application context, by extending existing ideas which have been developed for autonomous problems. Defect-based local error indicators appear to be a reasonable choice, but other techniques - if applicable - may be competitive.

# Literature

- W.Auzinger, O. Koch and M. Thalhammer, Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part II: Higher-order methods for linear problems, J. Comput. Appl. Math. 255, 384-403 (2014) DOI:10.1016/j.cam.2013.04.043
- W. Auzinger, H. Hofstätter, O. Koch and M. Thalhammer, Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case, J. Comput. Appl. Math. 273, 182-204 (2015) DOI:10.1016/j.cam.2014.06.012
- W. Auzinger, O. Koch and M. Thalhammer, Defect-based local error estimators for high-order splitting methods involving three linear operators, Numer. Algor. 70, 61-91 (2015) DOI:10.1007/s11075-014-9935-8
- W. Auzinger, T. Kassebacher, O. Koch and M. Thalhammer, Adaptive splitting methods for nonlinear Schrödinger equations in the semiclassical regime, Numer. Algor. 72, 1-35 (2016) DOI:10.1007/s11075-015-0032-4
- W. Auzinger, O. Koch and E. Weinmüller. Analysis of a new error estimate for collocation methods applied to singular boundary value problems, SIAM J. Numer. Anal. 42, 2366-2386 (2005) DOI:10.1137/S0036142902418928