Splitting methods for separable Hamiltonians

A widely used class of symplectic integrators is formed by the splitting methods.

Assume that the Hamiltonian is separable, meaning that it can be written in the form

${\displaystyle H(p,q)=T(p)+V(q).}$

(1)

This happens frequently in Hamiltonian mechanics, with T being the kinetic energy and V the potential energy.

For the notational simplicity, let us introduce the symbol ${\displaystyle z=(q,p)}$ to denote the canonical coordinates including both the position and momentum coordinates. Then, the set of the Hamilton's equations given in the introduction can be expressed in a single expression as

${\displaystyle {\dot {z}}=\{z,H(z)\},}$

(2)

where ${\displaystyle \{\cdot ,\cdot \}}$ is a Poisson bracket. Furthermore, by introducing an operator ${\displaystyle D_{H}\cdot =\{\cdot ,H\}}$, which returns a Poisson bracket of the operand with the Hamiltonian, the expression of the Hamilton's equation can be further simplified to

${\displaystyle {\dot {z}}=D_{H}z.}$

The formal solution of this set of equations is given as a matrix exponential:

${\displaystyle z(\tau )=\exp(\tau D_{H})z(0).}$

(3)

Note the positivity of ${\displaystyle \tau D_{H}}$ in the matrix exponential.

When the Hamiltonian has the form of equation (1), the solution (3) is equivalent to

${\displaystyle z(\tau )=\exp[\tau (D_{T}+D_{V})]z(0).}$

(4)

The SI scheme approximates the time-evolution operator ${\displaystyle \exp[\tau (D_{T}+D_{V})]}$ in the formal solution (4) by a product of operators as

${\displaystyle {\begin{aligned}\exp[\tau (D_{T}+D_{V})]&=\prod _{i=1}^{k}\exp(c_{i}\tau D_{T})\exp(d_{i}\tau D_{V})+O(\tau ^{k+1})\\&=\exp(c_{1}\tau D_{T})\exp(d_{1}\tau D_{V})\dots \exp(c_{k}\tau D_{T})\exp(d_{k}\tau D_{V})+O(\tau ^{k+1}),\end{aligned}}}$

(5)

where ${\displaystyle c_{i}}$ and ${\displaystyle d_{i}}$ are real numbers, ${\displaystyle k}$ is an integer, which is called the order of the integrator, and where ${\textstyle \sum _{i=1}^{k}c_{i}=\sum _{i=1}^{k}d_{i}=1}$. Note that each of the operators ${\displaystyle \exp(c_{i}\tau D_{T})}$ and ${\displaystyle \exp(d_{i}\tau D_{V})}$ provides a symplectic map, so their product appearing in the right-hand side of (5) also constitutes a symplectic map.

Since ${\displaystyle D_{T}^{2}z=\{\{z,T\},T\}=\{({\dot {q}},0),T\}=(0,0)}$ for all ${\displaystyle z}$, we can conclude that

${\displaystyle D_{T}^{2}=0.}$

(6)

By using a Taylor series, ${\displaystyle \exp(aD_{T})}$ can be expressed as

${\displaystyle \exp(aD_{T})=\sum _{n=0}^{\infty }{\frac {(aD_{T})^{n}}{n!}},}$

(7)

where ${\displaystyle a}$ is an arbitrary real number. Combining (6) and (7), and by using the same reasoning for ${\displaystyle D_{V}}$ as we have used for ${\displaystyle D_{T}}$, we get

${\displaystyle {\begin{cases}\exp(aD_{T})&=1+aD_{T},\\\exp(aD_{V})&=1+aD_{V}.\end{cases}}}$

(8)

In concrete terms, ${\displaystyle \exp(c_{i}\tau D_{T})}$ gives the mapping

${\displaystyle {\begin{pmatrix}q\\p\end{pmatrix}}\mapsto {\begin{pmatrix}q+\tau c_{i}{\frac {\partial T}{\partial p}}(p)\\p\end{pmatrix}},}$

and ${\displaystyle \exp(d_{i}\tau D_{V})}$ gives

${\displaystyle {\begin{pmatrix}q\\p\end{pmatrix}}\mapsto {\begin{pmatrix}q\\p-\tau d_{i}{\frac {\partial V}{\partial q}}(q)\\\end{pmatrix}}.}$

Note that both of these maps are practically computable.

Splitting methods for general nonseparable Hamiltonians

General nonseparable Hamiltonians can also be explicitly and symplectically integrated.

To do so, Tao introduced a restraint that binds two copies of phase space together to enable an explicit splitting of such systems.^[7] The idea is, instead of ${\displaystyle H(Q,P)}$, one simulates ${\displaystyle {\bar {H}}(q,p,x,y)=H(q,y)+H(x,p)+\omega \left(\left\|q-x\right\|_{2}^{2}/2+\left\|p-y\right\|_{2}^{2}/2\right)}$, whose solution agrees with that of ${\displaystyle H(Q,P)}$ in the sense that ${\displaystyle q(t)=x(t)=Q(t),p(t)=y(t)=P(t)}$.

The new Hamiltonian is advantageous for explicit symplectic integration, because it can be split into the sum of three sub-Hamiltonians, ${\displaystyle H_{A}=H(q,y)}$, ${\displaystyle H_{B}=H(x,p)}$, and ${\displaystyle H_{C}=\omega \left(\left\|q-x\right\|_{2}^{2}/2+\left\|p-y\right\|_{2}^{2}/2\right)}$. Exact solutions of all three sub-Hamiltonians can be explicitly obtained: both ${\displaystyle H_{A},H_{B}}$ solutions correspond to shifts of mismatched position and momentum, and ${\displaystyle H_{C}}$ corresponds to a linear transformation. To symplectically simulate the system, one simply composes these solution maps.

In plasma physics

In recent decades symplectic integrator in plasma physics has become an active research topic,^[8] because straightforward applications of the standard symplectic methods do not suit the need of large-scale plasma simulations enabled by the peta- to exa-scale computing hardware. Special symplectic algorithms need to be customarily designed, tapping into the special structures of physics problem under investigation. One such example is the charged particle dynamics in an electromagnetic field. With the canonical symplectic structure, the Hamiltonian of the dynamics is

$${\displaystyle H({\boldsymbol {p}},{\boldsymbol {x}})={\frac {1}{2}}\left({\boldsymbol {p}}-{\boldsymbol {A}}\right)^{2}+\phi ,}$$

whose ${\textstyle {\boldsymbol {p}}}$-dependence and ${\textstyle {\boldsymbol {x}}}$-dependence are not separable, and standard explicit symplectic methods do not apply. For large-scale simulations on massively parallel clusters, however, explicit methods are preferred.

To overcome this difficulty, we can explore the specific way that the ${\textstyle {\boldsymbol {p}}}$-dependence and ${\textstyle {\boldsymbol {x}}}$-dependence are entangled in this Hamiltonian, and try to design a symplectic algorithm just for this or this type of problem. First, we note that the ${\textstyle {\boldsymbol {p}}}$-dependence is quadratic, therefore the first order symplectic Euler method implicit in ${\textstyle {\boldsymbol {p}}}$ is actually explicit. This is what is used in the canonical symplectic particle-in-cell (PIC) algorithm.^[9] To build high order explicit methods, we further note that the ${\textstyle {\boldsymbol {p}}}$-dependence and ${\textstyle {\boldsymbol {x}}}$-dependence in this ${\textstyle H({\boldsymbol {p}},{\boldsymbol {x}})}$ are product-separable, 2nd and 3rd order explicit symplectic algorithms can be constructed using generating functions,^[10] and arbitrarily high-order explicit symplectic integrators for time-dependent electromagnetic fields can also be constructed using Runge-Kutta techniques.^[11]

A more elegant and versatile alternative is to look at the following non-canonical symplectic structure of the problem,

$${\displaystyle i_{({\dot {\boldsymbol {x}}},{\dot {\boldsymbol {v}}})}\Omega =-dH,\ \ \ \Omega =d({\boldsymbol {v}}+{\boldsymbol {A}})\wedge d{\boldsymbol {x}},\ \ \ H={\frac {1}{2}}{\boldsymbol {v}}^{2}+\phi .}$$

Here ${\textstyle \Omega }$ is a non-constant non-canonical symplectic form. General symplectic integrator for non-constant non-canonical symplectic structure, explicit or implicit, is not known to exist. However, for this specific problem, a family of high-order explicit non-canonical symplectic integrators can be constructed using the He splitting method.^[12] Splitting ${\textstyle H}$ into 4 parts,

$${\displaystyle {\begin{aligned}H&=H_{x}+H_{y}+H_{z}+H_{\phi },\\H_{x}&={\frac {1}{2}}v_{x}^{2},\ \ H_{y}={\frac {1}{2}}v_{y}^{2},\ \ H_{z}={\frac {1}{2}}v_{z}^{2},\ \ H_{\phi }=\phi ,\end{aligned}}}$$

we find serendipitously that for each subsystem, e.g.,

$${\displaystyle i_{({\dot {\boldsymbol {x}}},{\dot {\boldsymbol {v}}})}\Omega =-dH_{x}}$$

and

$${\displaystyle i_{({\dot {\boldsymbol {x}}},{\dot {\boldsymbol {v}}})}\Omega =-dH_{\phi },}$$

the solution map can be written down explicitly and calculated exactly. Then explicit high-order non-canonical symplectic algorithms can be constructed using different compositions. Let ${\textstyle \Theta _{x},\Theta _{y},\Theta _{z}}$ and ${\textstyle \Theta _{\phi }}$ denote the exact solution maps for the 4 subsystems. A 1st-order symplectic scheme is

$${\displaystyle {\begin{aligned}\Theta _{1}\left(\Delta \tau \right)=\Theta _{x}\left(\Delta \tau \right)\Theta _{y}\left(\Delta \tau \right)\Theta _{z}\left(\Delta \tau \right)\Theta _{\phi }\left(\Delta \tau \right)~.\end{aligned}}}$$

A symmetric 2nd-order symplectic scheme is,

$${\displaystyle {\begin{aligned}\Theta _{2}\left(\Delta \tau \right)&=\Theta _{x}\left(\Delta \tau /2\right)\Theta _{y}\left(\Delta \tau /2\right)\Theta _{z}\left(\Delta \tau /2\right)\Theta _{\phi }\left(\Delta \tau \right)\\&\Theta _{z}\left(\Delta t/2\right)\Theta _{y}\left(\Delta t/2\right)\Theta _{x}\left(\Delta t/2\right)\!,\end{aligned}}}$$

which is a customarily modified Strang splitting. A ${\textstyle 2(l+1)}$-th order scheme can be constructed from a ${\textstyle 2l}$-th order scheme using the method of triple jump,

$${\displaystyle {\begin{aligned}\Theta _{2(l+1)}(\Delta \tau )&=\Theta _{2l}(\alpha _{l}\Delta \tau )\Theta _{2l}(\beta _{l}\Delta \tau )\Theta _{2l}(\alpha _{l}\Delta \tau )~,\\\alpha _{l}&=1/(2-2^{1/(2l+1)})~,\\\beta _{l}&=1-2\alpha _{l}~.\end{aligned}}}$$

The He splitting method is one of key techniques used in the structure-preserving geometric particle-in-cell (PIC) algorithms.^{[13][14][15][16]}