Variational Partitioned Runge-Kutta Integrators

Variational partitioned Runge-Kutta methods solve Lagranian systems in implicit form, i.e.,

\[\begin{aligned} p &= \dfrac{\partial L}{\partial \dot{q}} (q, \dot{q}) , & \dot{p} &= \dfrac{\partial L}{\partial q} (q, \dot{q}) , \end{aligned}\]

by the following scheme,

\[\begin{aligned} P_{n,i} &= \dfrac{\partial L}{\partial \dot{q}} (Q_{n,i}, V_{n,i}) , & F_{n,i} &= \dfrac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , \\ Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} \, V_{n,j} , & P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \bar{a}_{ij} \, F_{n,j} , \\ q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} \, V_{n,i} , & p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{b}_{i} \, F_{n,i} . \end{aligned}\]

Here, $s$ denotes the number of internal stages, $a_{ij}$ and $\bar{a}_{ij}$ are the coefficients of the Runge-Kutta method and $b_{i}$ and $\bar{b}_{i}$ the corresponding weights. If the coefficients satisfy the symplecticity conditions,

\[\begin{aligned} b_{i} \bar{a}_{ij} + \bar{b}_{j} a_{ji} &= b_{i} \bar{b}_{j} & & \text{and} & \bar{b}_{i} &= b_{i} , \end{aligned}\]

these methods correspond to the position-momentum form of the discrete Lagrangian [Matthew Marsden Jerrold E. AND West (2001)]

\[L_{d} (q_{n}, q_{n+1}) = h \sum \limits_{i=1}^{s} b_{i} \, L \big( Q_{n,i}, V_{n,i} \big) .\]

While these integrators show favourable properties for systems with regular Lagrangian, they are usually not applicable for degenerate Lagrangian systems, in particular those with Lagrangians of the form $L (q, \dot{q}) = \vartheta(q) \cdot \dot{q} - H(q)$. While variational integrators are still applicable in the case of $\vartheta$ being a linear function of $q$, they are often found to be unstable when $\vartheta$ is a nonlinear function of $q$ as is the case with Lotka-Volterra systems, guiding centre dynamics and various nonlinear oscillators. To mitigate this problem, projection methods have been developed, which when applied to variational integrators provide long-time stable integrators for general degenerate Lagrangian systems that maintain conservation of energy and momenta [Michael Kraus (2017)].

GeometricIntegrators.jl provides the following VPRK methods (some are still experimental):

For testing purposes IntegratorVPRKpStandard provides some additional constructors (these methods are generally unstable):

All of the above integrators are applied to an IODE.