Closed-form modified Hamiltonians for integrable numerical integration schemes
For researchers in geometric numerical integration, this work provides rare convergent examples that may improve understanding of long-time accuracy of symplectic schemes.
The paper constructs explicit examples of nonlinear Hamiltonian systems where the modified Hamiltonian converges and has a closed-form expression, using discrete integrable systems. These examples demonstrate that integrable mappings can provide convergent modified Hamiltonians, contrasting with typical divergent series.
Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually diverges. In contrast, this paper constructs and analyzes explicit examples of nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. We present cases of one- and two-degrees symplectic mappings arising as reductions of nonlinear integrable lattice equations, for which the modified Hamiltonians can be computed in closed form. These modified Hamiltonians are also given as power series in the time step by Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example displays an implicit dependence on the time step which could be of relevance to certain implicit schemes in numerical analysis. In the light of these examples, the potential importance of integrable mappings to the field of geometric numerical integration is discussed.