Geometric Integration: Symplectic, Variational, and Structure-Preserving¶
Job: develop the symplectic methods (Stormer–Verlet), the variational integrators (Marsden–West), the KAM / backward error (Benettin–Giorgilli), and the volume-preserving / Liouville — with the Stormer–Verlet / backward error as the worked example.
Core¶
The symplectic methods and the Hamiltonian¶
| Hamiltonian / symplectic / Stormer–Verlet | Description |
|---|---|
| Hamiltonian \(H(q,p)\) | the flow the symplectic (\(dp\wedge dq\) preserved, the dynamical systems / Hamiltonian) |
| Symplectic integrator | the Stormer–Verlet / the symplectic RK (the \(dp\wedge dq\) preserved) |
| Backward error | the modified \(\tilde H=H+h^2H_2+\dots\) (the exponentially small drift) |
The variational integrators¶
| Variational / \(L_d\) / discrete EL / Marsden–West | Description |
|---|---|
| Discrete Lagrangian \(L_d(q_0,q_1)\) | the discrete action \(\sum L_d(q_k,q_{k+1})\) |
| Discrete Euler–Lagrange | the \(\frac{\partial L_d}{\partial q_k}+\frac{\partial L_d}{\partial q_{k-1}}=0\) (the discrete EL) |
| Variational integrator | the Marsden–West (the variational, the symplectic the consequence) |
The KAM and the long-time stability¶
| KAM / long-time / backward error / Benettin–Giorgilli | Description |
|---|---|
| KAM | the near-integrable the invariant tori (the dynamical systems / Hamiltonian) |
| Long-time / backward error | the modified equation, the exponentially small drift (the Benettin–Giorgilli) |
The volume-preserving and the Liouville¶
| Volume / Liouville / integrator | Description |
|---|---|
| Volume / Liouville | the divergence-free flow, the phase space volume (the ergodic theory) |
| Volume-preserving integrator | the divergence-free (the ergodic theory) |
See it¶
Stormer–Verlet for the harmonic oscillator and the backward error / modified equation. The harmonic oscillator \(H=\frac12(p^2+q^2)\), the Stormer–Verlet the symplectic (the \(dp\wedge dq\) preserved, the dynamical systems / Hamiltonian). The backward error: the \(\tilde H=H+h^2H_2+\dots\) the modified (the exponentially small the drift, the Benettin–Giorgilli the long-time). The key point: the Stormer–Verlet the symplectic, the backward error the modified / the long-time stable.
Check¶
Show that the Stormer–Verlet is symplectic, and explain the backward error / modified Hamiltonian.
The Stormer–Verlet: \(p_{n+1/2}=p_n-\frac{h}{2}\nabla V(q_n)\), \(q_{n+1}=q_n+h p_{n+1/2}\), \(p_{n+1}=p_{n+1/2}-\frac{h}{2}\nabla V(q_{n+1})\). The symplectic: the \(dp_{n+1}\wedge dq_{n+1}=dp_n\wedge dq_n\) (the \(dp\wedge dq\) preserved, the Stormer–Verlet the symplectic). The backward error: the \(\Phi_h^V=\Phi_{\tilde H}^{h}\) the exact flow of the modified \(\tilde H=H+h^2H_2+h^4H_4+\dots\) (the formal series, the exponentially small the drift, the Benettin–Giorgilli the truncation the exponentially small). So the Stormer–Verlet the symplectic, the backward error the modified / the long-time.
Explain the variational integrator and the discrete Euler–Lagrange.
The discrete Lagrangian: \(L_d(q_0,q_1)\approx\int_0^h L(q,\dot q)\,dt\) (the quadrature, the discrete action \(\sum L_d(q_k,q_{k+1})\)). The discrete EL: the \(\frac{\partial L_d}{\partial q_k}(q_{k-1},q_k)+\frac{\partial L_d}{\partial q_k}(q_k,q_{k+1})=0\) (the discrete Euler–Lagrange, the variational). The variational integrator: the Marsden–West (the discrete EL \(\Rightarrow\) the symplectic, the variational the symplectic the consequence). So the variational the \(L_d\), the discrete EL, the symplectic.
Pitfalls¶
- The symplectic the \(dp\wedge dq\) (the preserved, the dynamical systems / Hamiltonian) — the symplectic the structure, the non-symplectic the drift.
- The backward error the modified (\(\tilde H=H+h^2H_2+\dots\), the exponentially small) — the backward error the formal, the Benettin–Giorgilli the truncation.
- The variational the \(L_d\) (the discrete Lagrangian, the Marsden–West) — the variational the symplectic, the discrete EL.
- The volume / Liouville (the divergence-free, the ergodic theory) — the volume the structure, the Liouville the preserved.
Refs¶
- Hairer–Lubich–Wanner, Geometric Numerical Integration (2006); the symplectic / the backward error
- Marsden–West, "Discrete mechanics and variational integrators" (2001); the variational
- Benettin–Giorgilli, "On the Hamiltonian interpolation of near-the-identity maps" (1994); the exponentially small
- Leimkuhler–Reich, Simulating Hamiltonian Dynamics (2005); the geometric integration
Next¶
NA index · Foundations · FEM & Variational · Iterative & Multigrid · High-Dimensional & UQ · Dynamical systems / Hamiltonian · Ergodic theory