Geometric numerical integration

Let $Q\subseteq {\mathbb{R}}^d$ be open, let $V\in C^1(Q,\mathbb{R})$, and let $M$ be a real symmetric positive-definite matrix. Use the force-kick $K_a$ and drift $D_b$ already defined in One-step leapfrog map as force and drift shears, with $U$ there replaced by $V$. The local kick is the restriction of $K_a$ to $Q\times {\mathbb{R}}^d$. The local drift is the restriction of $D_b$ to $\{(q,p):q\in Q,q+b{M}^{-1}p\in Q\}\mathrm{.}$

For a signed step $h$, the local leapfrog map is the restriction of $K_{h\mathrm{/}2}\circ D_h\circ K_{h\mathrm{/}2}$ to the states for which all three intermediate configurations lie in $Q$. Its output is defined only on that admitted set. This definition changes the configuration domain of the published map; it does not redefine the global leapfrog map or assert that the local map is complete.

Let $U\subseteq {\mathbb{R}}^{2d}$ be open, $H\in C^{\mathrm{\infty }}(U)$, and use the canonical symplectic form $\omega ={\sum }_{i=1}^{d}d{q}_i\wedge d{p}_i$ with matrix $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$. Define $X_H$ by ${\iota }_{X_H}\omega =dH$. Let ${\phi }_t$ be the local flow on any open set where it is defined. At each initial state and time for which the intervening trajectory lies in $U$, its derivative satisfies $D{\phi }_t(z{)}^{T}JD{\phi }_t(z)=J\mathrm{.}$ This is a local, choice-free variational proof; no completeness is asserted.

Input an admitted state $(q,p)\in Q\times {\mathbb{R}}^d$, step $h>0$, $V\in C^1(Q)$, and symmetric positive-definite $M$. First set $p^{+}=p-h\mathrm{\nabla }V(q)$. Then set $q^{+}=q+h{M}^{-1}{p}^{+}$. If $q^{+}\mathrm{\notin }Q$, return domain-failure; otherwise return $(q^{+},p^{+})$. This fixes the kick-then-drift ordering. The returned map is the partial composition $D_h\circ K_h$ in the notation of Local domain restriction for kick, drift, and leapfrog maps.

For exact mathematical statements the displayed operations are evaluated exactly. A finite implementation that cannot certify the state and force evaluation returns inconclusive; it does not report a step outside $Q$ as successful.

Let $d\ge 1$, let $U\subseteq {\mathbb{R}}^{2d}$ be open, and let $H\in C^2(U)$. Write $z=(q,p)$ in canonical coordinates, set $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, and use the Euclidean norm and its induced operator norm. Define the finite-regularity coordinate field $F(q,p):=({\mathrm{\nabla }}_{p}H(q,p),-{\mathrm{\nabla }}_{q}H(q,p))=J\mathrm{\nabla }H(q,p)\mathrm{.}$

Given $z\in U$ and a signed step $h$, supply a candidate radius $\rho \ge 0$. If no radius is supplied or $\rho <0$, return branch-not-certified. If the closed ball $\bar{B}(z,\rho )$ is certified not to lie in $U$, return domain-failure; if its containment in $U$ cannot be certified, return branch-not-certified. For a ball contained in $U$, supply certified bounds $M,L\ge 0$ with $M\ge {\sup }_{W\in \bar{B}(z,\rho )}\mathrm{\parallel }F(W)\mathrm{\parallel },L\ge {\sup }_{W\in \bar{B}(z,\rho )}\mathrm{\parallel }DF(W)\mathrm{\parallel }\mathrm{.}$ If either bound or its certificate is unavailable, or if either admissibility inequality $\mathrm{\mid }h\mathrm{\mid }M\mathrm{/}2\le \rho$ or $\mathrm{\mid }h\mathrm{\mid }L\mathrm{/}2<1$ fails, return branch-not-certified. Otherwise the exact local implicit midpoint step selects the unique $Z\in \bar{B}(z,\rho )$ satisfying $Z=z+\frac{h}{2}F(Z)$ and returns $z^{+}=2Z-z$. The returned state is in ${\mathbb{R}}^{2d}$; a further step is admitted only when that state lies in $U$ and passes a new ball check.

For a finite iteration implementation of an admitted branch, input an output tolerance $\tau >0$ and an update budget $B\in {\mathbb{N}}_0$. Use $\theta =\mathrm{\mid }h\mathrm{\mid }L\mathrm{/}2$ and proceed as follows.

1. Set $Z_0=z$. If $B=0$, return inconclusive with current stage iterate $Z_0$, approximate output $z$, and output-error bound $2\rho$.
2. If $B\ge 1$, compute $Z_1=z+(h\mathrm{/}2)F(Z_0)$ and set $d_1=\mathrm{\parallel }{Z}_1-Z_0\mathrm{\parallel }$.
3. For $k=1,\dots ,B$, set $b_k={\theta }^k{d}_1\mathrm{/}(1-\theta )$ and the output radius $r_k=2b_k$. If $r_k\le \tau$, return the approximation $2Z_k-z$ with output-error certificate $r_k$. Otherwise, if $k<B$, compute $Z_{k+1}=z+(h\mathrm{/}2)F(Z_k)$ and continue. If the check fails at $k=B$, return inconclusive with current stage iterate $Z_B$, approximate output $2Z_B-z$, and output-error bound $2b_B$.

When $\theta =0$, use ${\theta }^k=0$ for $k\ge 1$; the procedure never evaluates $0^0$. If a finite implementation cannot certify an input or an exact operation needed for these instructions, return inconclusive without a tolerance claim. The certificates concern the displayed exact recurrence and exact output operations; inexact evaluations require additional certified error accounting. An approximate result is not asserted to be exactly symplectic. The exact map is the selected branch value $2Z-z$.

Let $Q\subseteq {\mathbb{R}}^d$ be open, $V\in C^2(Q)$, and $M$ be symmetric positive definite. Put $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$. At every point of their open domains, the local kick and drift from Local domain restriction for kick, drift, and leapfrog maps are $C^1$ and their Jacobians $A$ satisfy $A^{T}JA=J$. Here this matrix identity is the finite-regularity meaning of symplecticity; no smooth-manifold definition is being applied to a map of insufficient regularity.

Under the hypotheses of Canonical kick and drift shears satisfy the symplectic Jacobian identity, every admitted local leapfrog step and every admitted kick-then-drift symplectic Euler step is a $C^1$ map with continuous Jacobian satisfying $D{\mathrm{\Phi }}^{T}JD\mathrm{\Phi }=J$ on its open admitted domain. The result concerns symplecticity, not energy conservation.

Let $d\ge 1$, $Q\subseteq {\mathbb{R}}^d$ be open, $V\in C^3(Q)$, and $M$ symmetric positive definite. Set $H(q,p)=\frac{1}{2}{p}^T{M}^{-1}p+V(q)$ and $F(q,p)=(M^{-1}p,-\mathrm{\nabla }V(q))$. Fix $T>0$ and $h=T\mathrm{/}N$ with $N\ge 1$. Suppose the exact solution, numerical states, all kick/drift intermediate states, and every line segment used below lie in one compact convex tube $K⋐Q\times {\mathbb{R}}^d$, with the derivatives of $F$ through order two bounded there. For $z_0$ in that tube, write $e_n^E$ for the error of iterated kick-then-drift symplectic Euler and $e_n^V$ for the error of iterated local leapfrog relative to the exact solution at $t_n=nh$. Write ${\mathrm{\Psi }}_h^E$ and ${\mathrm{\Psi }}_h^V$ for their admitted one-step maps and ${\phi }_h$ for the exact flow. For sufficiently small $h$, constants $C_E,C_V,L$ independent of $h,N,n$ satisfy $\mathrm{\parallel }{\mathrm{\Psi }}_h^E(z)-{\phi }_h(z)\mathrm{\parallel }\le C_E{h}^2,\mathrm{\parallel }{\mathrm{\Psi }}_h^V(z)-{\phi }_h(z)\mathrm{\parallel }\le C_V{h}^3$ for admitted $z\in K$, and whenever $nh\le T$, $e_n^E\le e^{LT}{e}_0+C_{E}T{e}^{LT}h,e_n^V\le e^{LT}{e}_0+C_{V}T{e}^{LT}{h}^2\mathrm{.}$ Thus symplectic Euler has global order one and leapfrog has global order two on the stated controlled tube. No unconditional domain-admissibility claim is made.

Let $d\ge 1$, $U\subseteq {\mathbb{R}}^{2d}$ be open, $H\in C^2(U)$, $z=(q,p)$, and $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$. Use the Euclidean norm and its induced operator norm, and define $F(q,p)=({\mathrm{\nabla }}_{p}H(q,p),-{\mathrm{\nabla }}_{q}H(q,p))=J\mathrm{\nabla }H(q,p)$. For a signed step $h$, suppose a declared closed ball $\bar{B}(z,\rho )\subset U$, with $\rho \ge 0$, has certified bounds $\mathrm{\parallel }F\mathrm{\parallel }\le M$ and $\mathrm{\parallel }DF\mathrm{\parallel }\le L$ satisfying $\mathrm{\mid }h\mathrm{\mid }M\mathrm{/}2\le \rho$ and $\theta :=\mathrm{\mid }h\mathrm{\mid }L\mathrm{/}2<1$. The exact implicit midpoint branch Local implicit midpoint step for a Hamiltonian vector field then has a unique stage root $Z$ in this ball. The exact iterates $Z_0=z$, $Z_{k+1}=z+(h\mathrm{/}2)F(Z_k)$ stay in the ball, converge to $Z$, and for every $k\in {\mathbb{N}}_0$ satisfy $\mathrm{\parallel }{Z}_k-Z\mathrm{\parallel }\le b_k:=\frac{{\theta }^k\mathrm{\parallel }{Z}_1-Z_0\mathrm{\parallel }}{1-\theta },\mathrm{\parallel }(2Z_k-z)-(2Z-z)\mathrm{\parallel }\le 2b_k\mathrm{.}$ Here ${\theta }^0=1$, including when $\theta =0$; if $\theta =0$, the first update equals $Z$ and $b_k=0$ for $k\ge 1$. Also $\mathrm{\parallel }(2Z-z)-z\mathrm{\parallel }\le 2\rho$, which supplies an output-error bound when no update is computed.

For every finite tolerance $\tau >0$ and budget $B\in {\mathbb{N}}_0$, the implementation in Local implicit midpoint step for a Hamiltonian vector field terminates after at most $B$ updates and at most $B$ evaluations of $F$; each reported error radius is a valid bound.

Assume additionally $H\in C^3(U)$ and that these ball hypotheses hold uniformly for initial values $z$ in a compact convex tube for all $\mathrm{\mid }h\mathrm{\mid }\le h_0$, with some $h_0>0$ and common $\rho ,M,L$, where $h_{0}L\mathrm{/}2\le 1\mathrm{/}2$. At every admitted initial value the selected branch is locally the restriction of a $C^1$ solution branch, and its exact local map ${\mathrm{\Psi }}_h(z)=2Z-z$ satisfies $D{\mathrm{\Psi }}_h(z{)}^{T}JD{\mathrm{\Psi }}_h(z)=J$.

Fix $T>0$. Under these additional $C^3$ and uniform branch hypotheses, assume also that the exact and numerical trajectories, the branch balls, and the segments used in comparison lie in a compact convex subset of $U$ whenever $n\mathrm{\mid }h\mathrm{\mid }\le T$, with $F$ and its first two derivatives bounded there. For an initial value $z_0$ in the tube, define $z_{k+1}={\mathrm{\Psi }}_h(z_k)$ for $k\in {\mathbb{N}}_0$ and $e_n=\mathrm{\parallel }{z}_n-{\phi }_{nh}(z_0)\mathrm{\parallel }$ for $n\in {\mathbb{N}}_0$ (so $e_0=0$). Then for constants $C,L_1$ independent of $h,n$, every $z$ in the tube, and every $\mathrm{\mid }h\mathrm{\mid }\le h_0$ with $n\mathrm{\mid }h\mathrm{\mid }\le T$, $\mathrm{\parallel }{\mathrm{\Psi }}_h(z)-{\phi }_h(z)\mathrm{\parallel }\le C\mathrm{\mid }h{\mathrm{\mid }}^3,e_n\le e^{L_{1}T}{e}_0+CT{e}^{L_{1}T}\mathrm{\mid }h{\mathrm{\mid }}^2\mathrm{.}$ Here ${\phi }_h$ is the exact Hamiltonian flow. These symplecticity and trajectory-error conclusions concern the exact selected map; no such conclusion is asserted here for a finite iteration output.

Let $Q\subseteq {\mathbb{R}}^d$ be open and star-shaped about a point $q_{\ast }\in Q$, meaning $q_{\ast }+t(q-q_{\ast })\in Q$ for every $q\in Q$ and $t\in [0,1]$. Let $M$ be real symmetric positive-definite, $U=Q\times {\mathbb{R}}^d$, and equip $U$ with the canonical form $\omega ={\sum }_{i=1}^{d}d{q}_i\wedge d{p}_i$ and matrix $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$. Set $H(q,p)=\frac{1}{2}{p}^T{M}^{-1}p+V(q)$ with $V\in C^{\mathrm{\infty }}(Q)$. Let ${\mathrm{\Phi }}_h$ be the kick-then-drift symplectic-Euler map, the local leapfrog map, or an admitted implicit-midpoint branch, on a compact subset where the chosen map is defined. Write $r=1$ for the kick-then-drift symplectic-Euler map and $r=2$ for leapfrog or implicit midpoint, as proved in Local and finite-time trajectory error for symplectic Euler and leapfrog and The admitted implicit midpoint branch is symplectic and second order. For a smooth test function $f$, define the pullback $P_{h}f=f\circ {\mathrm{\Phi }}_h$ by its Taylor coefficients at $h=0$. For fixed finite $N\ge r$, its truncated logarithm is the polynomial ${\log }_N(P_h)={\sum }_{k=1}^{N+1}\frac{(-1{)}^{k+1}}{k}(P_h-I{)}^k,$ computed modulo terms of order $h^{N+2}$. There are smooth vector fields $X_r,\dots ,X_N$ and globally defined functions $H_r,\dots ,H_N$ on $U$ such that $h^{-1}{\log }_N(P_h)={\mathcal{L}}_{X_H}+{\sum }_{j=r}^N{h}^j{\mathcal{L}}_{X_j}(\mathrm{m}\mathrm{o}\mathrm{d}{h}^{N+1}),{\iota }_{X_j}\omega =d{H}_j,$ where ${\mathcal{L}}_{X}f:=Df[X]$. Thus $H_h^{[N]}=H+{\sum }_{j=r}^N{h}^j{H}_j$ is a globally defined finite modified Hamiltonian. The logarithm is a finite formal polynomial for each fixed $N$; no convergence of an infinite series is asserted.