M. Oberguggenberger, A. Ostermann
Analysis for Computer Scientists. 2nd edition, Springer International Publishing (2018)
DOI 10.1007/978-3-319-91155-7
ISBN 978-3-319-91154-0

A. Ostermann, G. Wanner
Geometry by Its History. Springer-Verlag, Berlin Heidelberg (2012)
DOI 10.1007/978-3-642-29163-0
ISBN 978-3-642-29162-3 

A. Ostermann, F. Rousset, K. Schratz (2021): Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces
To appear in J. Eur. Math. Soc.

In this paper, we propose a new scheme for the integration of the periodic nonlinear Schrödinger equation and rigorously prove convergence rates at low regularity. The new integrator has decisive advantages over standard schemes at low regularity ... read more

M. Hochbruck, J. Leipold, A. Ostermann (2020): On the convergence of Lawson methods for semilinear stiff problems
Numerische Mathematik 145, 553-580

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a ... read more

L. Einkemmer (2020):  Semi-Lagrangian Vlasov simulation on GPUs
Comput. Phys. Commun., 254, 107351

In this paper, our goal is to efficiently solve the Vlasov equation on GPUs. A semi-Lagrangian discontinuous Galerkin scheme is used for the discretization. Such kinetic computations are extremely expensive due to the high-dimensional phase space. The SLDG code ... read more

M. Knöller, A. Ostermann, K. Schratz (2019): A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data
SIAM J. Numer. Anal. 57, 1967-1986

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrödinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be ... read more

L. Einkemmer (2019): A performance comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions
J. Comput. Phys., 376, 937–951

The purpose of the present paper is to compare two semi-Lagrangian methods in the context of the four-dimensional Vlasov–Poisson equation. More specifically, our goal is to compare the performance of the more recently developed semi-Lagrangian ... read more

A. Ostermann, K. Schratz (2018): Low regularity exponential-type integrators for semilinear Schrödinger equations
Found. Comput. Math. 18, 731-755

We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order ... read more

L. Einkemmer & C. Lubich (2018): A Low-Rank Projector-Splitting Integrator for the Vlasov–Poisson Equation
SIAM J. Sci. Comput., 40, B1330–B1360

Many problems encountered in plasma physics require a description by kinetic equations, which are posed in an up to six-dimensional phase space. A direct discretization of this phase space, often called the Eulerian approach, has many advantages but is ... read more

M. Caliari, P. Kandolf, A. Ostermann, S. Rainer (2016): The Leja method revisited: backward error analysis for the matrix exponential
SIAM J. Sci. Comput. 38, A1639-A1661

The Leja method is a polynomial interpolation procedure that can be used to compute matrix functions. In particular, computing the action of the matrix exponential on a given vector is a typical application. This quantity is required, e.g., in exponential ... read more

N. Crouseilles, L. Einkemmer & E. Faou (2015): A Hamiltonian splitting for the Vlasov–Maxwell system
J. Comput. Phys., 238, 224–240

A new splitting is proposed for solving the Vlasov–Maxwell system. This splitting is based on a decomposition of the Hamiltonian of the Vlasov–Maxwell system and allows for the construction of arbitrary high order methods by composition ... read more

L. Einkemmer, A. Ostermann (2015): Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions
SIAM J. Sci. Comput. 37, A1577-A1592

For diffusion-reaction equations employing a splitting procedure is attractive as it reduces the computational demand and facilitates a parallel implementation. Moreover, it opens up the possibility to construct second-order integrators that preserve positivity ... read more

L. Einkemmer, A. Ostermann (2014):  Convergence analysis of a discontinuous Galerkin / Strang splitting approximation for the Vlasov-Poisson equations
SIAM J. Numer. Anal. 52, 757-778

A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov--Poisson equations is provided. It is shown that under suitable assumptions the error is of order ... read more

V.T. Luan, A. Ostermann (2013): Exponential B-series: the stiff case
SIAM J. Numer. Anal. 51, 3431-3445

For the purpose of deriving the order conditions of exponential Runge--Kutta and exponential Rosenbrock methods, we extend the well-known concept of B-series to exponential integrators. As we are mainly interested in the stiff case, Taylor ... read more

M. Caliari, A. Ostermann, S. Rainer (2013): Meshfree exponential integrators
SIAM J. Sci. Comput. 35, A431-A452

For the numerical solution of time-dependent partial differential equations, a class of meshfree exponential integrators is proposed. These methods are of particular interest in situations where the solution of the differential equation concentrates on a small part of the computational domain which may vary in time. For the space discretization ... read more

M. Hochbruck, A. Ostermann (2010): Exponential integrators
Acta Numerica 19, 209-286

In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative ... read more