An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants. (English) Zbl 1100.65115
Using the formalism of B- and S-series, the authors characterise numerical integrators preserving quadratic and Hamiltonian invariants. They first provide alternative proofs of the following two already known results: a B-series integrator preserves quadratic invariants, if and only if it is a symplectic integrator, and it preserves Hamiltonian invariants, if and only if it satisfies certain algebraic conditions.
Then they introduce the notion of a modified invariant (a formal series which is \(O(h)\)-close to the given invariant) and prove the following three results: (i) a B-series integrator has a modified invariant for any quadratic invariant, if and only if it is conjugate to a symplectic integrator, (ii) it has a modified Hamiltonian invariant for a Hamiltonian problem, if and only if it is conjugate to a method preserving the Hamiltonian exactly, (iii) a symplectic B-series is formally conjugate to a B-series preserving the Hamiltonian exactly.
As a corollary one finds that the underlying one-step method of a symmetric multistep scheme is formally conjugate to a symplectic P-series when applied to Newton’s equations.
Then they introduce the notion of a modified invariant (a formal series which is \(O(h)\)-close to the given invariant) and prove the following three results: (i) a B-series integrator has a modified invariant for any quadratic invariant, if and only if it is conjugate to a symplectic integrator, (ii) it has a modified Hamiltonian invariant for a Hamiltonian problem, if and only if it is conjugate to a method preserving the Hamiltonian exactly, (iii) a symplectic B-series is formally conjugate to a B-series preserving the Hamiltonian exactly.
As a corollary one finds that the underlying one-step method of a symmetric multistep scheme is formally conjugate to a symplectic P-series when applied to Newton’s equations.
Reviewer: Werner M. Seiler (Kassel)
MSC:
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
Keywords:
preservation of invariants; symplectic integrator; B-series; S-series; Hamiltonian invariants; one-step method; Newton’s equationsReferences:
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