Fourth-order symplectic integration
Physica D
A numerical algorithm for Hamiltonian systems
Journal of Computational Physics
On the numerical integration of ordinary differential equations by symmetric composition methods
SIAM Journal on Scientific Computing
Nth-order operator splitting schemes and nonreversible systems
SIAM Journal on Numerical Analysis
Explicit Symplectic Integrators Using Hessian--Vector Products
SIAM Journal on Scientific Computing
Symplectic Integration with Processing: A General Study
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
On the Numerical Integration of Ordinary Differential Equations by Processed Methods
SIAM Journal on Numerical Analysis
Exact universality from any entangling gate without inverses
Quantum Information & Computation
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In this paper we analyse numerical integration methods applied to differential equations which are separable in solvable parts. These methods are compositions of flows associated with each part of the system. We propose an elementary proof of the necessary existence of negative coefficients if the schemes are of order, or effective order, p ≥ 3 and provide additional information about the distribution of these negative coefficients. It is shown that if the methods involve flows associated with more general terms this result does not necessarily apply and in some cases it is possible to build higher order schemes with positive coefficients.