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W. Auzinger, O. Koch, M. Thalhammer:
"Representation and estimation of the local error of higher-order exponential splitting schemes involving two or three sub-operators";
Talk: 12th International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2014), Rhodos (invited); 2014-09-22 - 2014-09-28; in: "Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014)", (2015), 150003.

We consider the numerical treatment of abstract evolution equations
\[
\partial_t\,u(t) = H u(t) = A\,u(t) + B\,u(t)\;[\,+\,\,\,C\,u(t)\,]\,, \quad u(0)~\text{given}\,,
\]
by higher-order exponential splitting schemes. The main focus is on linear problems.
%the nonlinear case is briefly commented on.
A single step of an exponential splitting scheme
with stepsize $ t $ and $ s $ sub-steps comprises a multiplicative composition of sub-flows
of the form
$
\mathcal{S}_j(t) = [\mathrm{e}^{t\,c_j\,C}]\,\,\mathrm{e}^{t\,b_j\,B}\,\mathrm{e}^{t\,a_j\,A}\,,
~j=1 \ldots s
$\,.
We present an algebraic theory of the structure of the local error. The leading term
is a linear combination of iterated commutators of the sub-operators $ A,\,B $
[and $ C $] involved. This fact can be exploited for the automatic setup of order conditions,
i.e., systems of polynomial equations for the coefficients $ a_j,\,b_j $ [and $ c_j $]
which have to be satisfied for a desired order $ p $.

In view of application to partial differential equations, an explicit, exact representation of the
local error is of interest. This can be obtained by performing a multiple variation-of-constants
integral expansion involving higher-order defect terms. The latter satisfy a multinomial
expansion, and the building blocks in this expansion are determined via yet another
recursively defined integral representation. We describe the rich combinatorial structure of
this local error expansion which is influenced by iterated commutators of the given sub-operators.
A defect-based a~posteriori local error estimator is also proposed.

linear evolution equations, high-order exponential operator splitting methods, local error, a posteriori local error estimation

http://dx.doi.org/10.1063/1.4912433

Project Head Othmar Koch:
Adaptives Splitting für nichtlineare Schrödingergleichungen