Integration methods

Lie Integrators

The following schemes solve systems of the form:

\[\begin{align} \dot{y} = A(y)y \end{align}\]

where \(A\) is a matrix. The solutions can be searched in a Lie group setting, if \(A(y)\) is a matrix-valued element of the corresponding Lie algebra.

Lie Forward Euler

The method can be expressed by the formula:

\[\begin{align} y_{n+1} = \exp{(hA(y_{n}))}\circ y_{n} \end{align}\]

where \(\exp\) and \(\circ\) are respectively the exponential map and the action on the considered Lie group, and \(h\) is the time step size.

Lie Backward Euler

Similar to the Lie Forward Euler, the Lie Backward Euler uses the operation on the considered Lie group to find the solution. Its scheme is:

\[\begin{align} y_{n+1} = \exp{(hA(y_{n+1}))}\circ y_{n} \end{align}\]

where, again, \(\exp\) and \(\circ\) are respectively the exponential map and the action on the considered Lie group, and \(h\) is the time step size.

Lie Midpoint Rule

Finally, we used for comparison also a Midpoint Rule in the Lie group setting. The expression for this method is:

\[\begin{align} y_{n+1} = \exp{(hA(\frac{y_{n+1}+y_{n}}{2}))}\circ y_{n} \end{align}\]

where \(\exp\) and \(\circ\) are respectively the exponential map and the action on the considered Lie group, and \(h\) is the time step size.