A Fundamental Theorem on the Structure of Symplectic Integrators

Abstract

I show that the basic structure of symplectic integrators is governed by a theorem which states {\it precisely}, how symplectic integrators with positive coefficients cannot be corrected beyond second order. All previous known results can now be derived quantitatively from this theorem. The theorem provided sharp bounds on second-order error coefficients explicitly in terms of factorization coefficients. By saturating these bounds, one can derive fourth-order algorithms analytically with arbitrary numbers of operators.

4 pages, no figures

References in corpus (8)

• Gradient Symplectic Algorithms for Solving the Schroedinger Equation with Time-Dependent Potentials
• On the construction of high-order force gradient algorithms for integration of motion in classical and quantum systems
• Path Integral Ground State with a Fourth-Order Propagator: Application to Condensed Helium
• Fourth-Order Algorithms for Solving the Imaginary Time Gross-Pitaevskii Equation in a Rotating Anisotropic Trap
• Quantum Statistical Calculations and Symplectic Corrector Algorithms
• Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions
• Short time evolved wave functions for solving quantum many-body problems
• The Complete Characterization of Fourth-Order Symplectic Integrators with Extended-Linear Coefficients

Cited by in corpus (5)

• Extrapolated High-Order Propagators for Path Integral Monte Carlo Simulations
• The inefficiency of re-weighted sampling and the curse of system size in high order path integration
• High-order Path Integral Monte Carlo methods for solving quantum dot problems
• The physics of symplectic integrators: perihelion advances and symplectic corrector algorithms
• Forward and non-forward symplectic integrators in solving classical dynamics problems