TY - JOUR
T1 - Explicit exactly energy-conserving methods for Hamiltonian systems
AU - Bilbao, Stefan
AU - Ducceschi, Michele
AU - Zama, Fabiana
N1 - Funding Information:
M. Ducceschi was supported by the European Research Council (ERC), under grant 2020-StG-950084-NEMUS . For the purpose of open access, the first author has applied a creative commons attribution (CC BY) licence to any author accepted manuscript version arising.
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as St{\"o}rmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.
AB - For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as St{\"o}rmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.
KW - Hamiltonian systems
KW - geometric numerical integration
KW - energy-conserving methods
KW - explicit methods
KW - finite difference methods
U2 - 10.1016/j.jcp.2022.111697
DO - 10.1016/j.jcp.2022.111697
M3 - Article
SN - 0021-9991
VL - 472
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 111697
ER -