TSBASICSYMPLECTIC

ODE solver using basic symplectic integration schemes https://en.wikipedia.org/wiki/Symplectic_integrator These methods are intended for separable Hamiltonian systems

\[ \begin{align*} \dot q &= \frac{dH(q,p,t)}{dp} \\ \dot p &= -\frac{dH(q,p,t)}{dq} \end{align*} \]

where the Hamiltonian can be split into the sum of kinetic energy and potential energy

\[ H(q,p,t) = T(p,t) + V(q,t). \]

As a result, the system can be generally represented by

\[ \begin{align*} \dot q &= f(p,t) = \frac{dT(p,t)}{dp} \\ \dot p &= g(q,t) = -\frac{dV(q,t)}{dq} \end{align*} \]

and solved iteratively with \(i \in [0, n]\)

\[ \begin{align*} q_{new} &= q_{old} + h d_i f(p_{old}, t_{old}) \\ t_{new} &= t_{old} + h d_i \\ p_{new} &= p_{old} + h c_i g(q_{new}, t_{new}) \end{align*} \]

The solution vector should contain both q and p, which correspond to (generalized) position and momentum respectively. Note that the momentum component could simply be velocity in some representations. The symplectic solver always expects a two-way splitting with the split names being “position” and “momentum”. Each split is associated with an IS object and a sub-TS that is intended to store the user-provided RHS function.

See Also

TS: Scalable ODE and DAE Solvers, TSCreate(), TSSetType(), TSRHSSplitSetIS(), TSRHSSplitSetRHSFunction(), TSType

Level

beginner

Location

src/ts/impls/symplectic/basicsymplectic/basicsymplectic.c

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Index of all TS routines
Table of Contents for all manual pages
Index of all manual pages