Coefficients of various splitting methods

Overview

This page gives a compendium of splitting methods both from the literature and newly constructed together with some properties relevant for assessing applicability and efficiency. A detailed description of the underlying algorithm and the methodology used to construct new schemes are given in Reference 1. Splitting into two, three, or four operators, respectively, is covered, and different types of pairs of schemes are given.

Splitting time integrators with s stages are defined for a linear evolution equation

∂t ψ =A ψ + B ψ [ + C ψ] ψ(0) = ψ0

(1)

as

S(h) u =  [eh ds Deh cs C]   eh bs B   eh as A ...   [eh c1 C]   eh b1 B   eh a1 A u

(2)

The construction works in an analogous way for nonlinear evolution equations by composing subflows.

The leading local error term for a splitting of order p has the form

hp+1/(p+1)! Σ1≤k≤Np κk Kk

(3)

where K_k contains all commutators of A,B,[C,D] applied to ψ up to order p. Splitting into two operators A,B fits into the general framework if we set c_i=d_i=0. To compare methods of equal order we consider

(Σk κk2)1/2

(4)

as a measure for the local error, with coefficients κ_k from (3). For the practical purpose of finding good methods we introduce a modified quantity LEM ('local error measure') which, for orders p>3, is an approximation for the quantity (4) and which can be computed directly from the order conditions. Its size is comparable to the size of (4).

For the purpose of local error estimation to serve as a basis for adaptive time stepping, several types of pairs of schemes have been designed:

1. "Emb": Pairs of embedded splitting formulae consist of two splitting schemes with a number of coinciding coefficients. For two operators, we choose a good higher order formula with coefficients a_i, b_i which serves as the error estimator and an associated lower order formula with coefficients a_lower,i, b_lower,i representing the basic integrator, and likewise for multi-operator splitting. By means of embedding, for the lower order formula we have selected the best one out of a set of candidates of equal computaional effort.
   
2. "Milne": An alternative method for local error estimation is the 'Milne device', based on a pair (I),(II) of schemes of equal order (here: p=2) with the property that κ_k,(1) = C · κ_k,(2) for all k, with some constant C.
   
3. "PP": A special case of adjoint pairs. For a scheme of odd order p, the leading term (3) of the local error changes its sign if the scheme is replaced by its adjoint. Therefore, the difference between these two steps divided by 2 is an efficient local error estimator which has also the full potential for parallelization. In a "PP" pair, the scheme has palindromic coefficients, and its adjoint is obtained by interchaning the roles of A and B. Pairs are explicitly mentioned here for which the underlying palindromic scheme is optimal among a set of comparable schemes, which has been observed in several cases.

In the menu on the left, a splitting method or pair can be selected to display its basic characterization, coefficients and LEM.

Several well-known and new schemes of orders up to p=6 are listed, but the list is not intended to be exhaustive. Also, note that for each scheme its adjoint has the same order. Furthermore, some more 'AB' schemes can be obtained from 'ABC schemes' by setting (e.g.) c_i=0, see, e.g., 'AKT 2-2 c'. This holds similarly for the 'ABCD' schemes.

Schemes which are part of a pair can also be used in a stand-alone way or be combined with another method for local error estimation.

Contact:

• Winfried Auzinger, Technische Universität Wien (w.auzinger@tuwien.ac.at)
• Othmar Koch, Universität Wien (othmar@othmar-koch.org)

Last update: January 24, 2024.