We introduce a defect correction principle for exponential operator splitting methods applied to time-dependent linear Schrödinger equations and construct a posteriori local error estimators for the Lie–Trotter and Strang splitting methods. Under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators, and along the way recover the kno...

The following text is a summary of the same-titled talk given within the frame of the seminar ’AKNUM: Seminar aus Numerik’ at the Vienna University of Technology. We address time splitting methods for the Schrödinger equation. In particular, we concentrate on the Strang splitting and prove second order convergence in the case of a bounded potential and given bounds on the (iterated) commutators...

A fractional time derivative is introduced into Burger’s equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equation...

In this work operator splitting techniques have been applied successfully to improve the accuracy of multi-scale Lithium-ion (Li-ion) battery models. A slightly simplified Li-ion battery model is derived, which can be solved on one time scale and multiple time scales. Different operator splitting schemes combined with different approximations are compared with the non-splitted reference solutio...

We study the solutions of Hermitian positive definite Toeplitz systems Tnx = b by the preconditioned conjugate gradient method. For preconditioner An the convergence rate is known to be governed by the distribution of the eigenvalues of the preconditioned matrix A−1 n Tn . New properties of the circulant preconditioners introduced by Strang, R. Chan, T. Chan, Szegö/Grenander and Tyrtyshnikov ar...

A newMatlab code for the generation of unstructured (3-node or 6-node) triangular meshes in two dimensions is described. The method is based on the Matlab mesh generator distmesh of Persson and Strang [SIAM Rev. 46:329-345, 2004]. As input, the code takes a signed distance function for the domain geometry. A mesh size function, for the spatial node distribution, is constructed using an approxim...

Multivariate quasi-projection operators Qj(f,φ,φ˜), associated with a function φ and distribution/function φ˜, are considered. The is supposed to satisfy the Strang-Fix conditions compatibility condition φ˜. Using technique based on Fourier multipliers, we study approximation properties of such for functions f from anisotropic Besov spaces Lp 1≤p≤∞. In particular, upper lower estimates Lp-error...

The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end the relevant notions and results of numerical analysis are presented, a variant of Chernoff’s product formula is proved and the general TrotterKato approximation theorem is used. The methods are applied t...

We provide a general abstract theory for the solvability and Galerkin approximation of nonlinear twofold saddle point problems. In particular, a Strang error estimate containing the consistency terms arising from the approximation of the continuous operators involved is deduced. Then we apply these results to analyse a fully discrete Galerkin scheme for a twofold saddle point formulation of a n...

We study the Strang splitting scheme for quasilinear Schrödinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schrödinger equations. Numerical tests a...