pyHamSys

pyHamSys is a Python package for scientific computing involving Hamiltonian systems

Installation:

pip install pyhamsys

Symplectic Integrators

pyHamSys includes a class SymplecticIntegrator containing the following symplectic splitting integrators:

• Verlet (order 2, all purpose), also referred to as Strang or Störmer-Verlet splitting
• From Forest, Ruth, Physica D 43, 105 (1990):
  • FR (order 4, all purpose)
• From Yoshida, Phys. Lett. A 150, 262 (1990):
  • Yo#: # should be replaced by an even integer, e.g., Yo6 for 6th order symplectic integrator (all purpose)
  • Yos6: (order 6, all purpose) optimized symplectic integrator (solution A from Table 1)
• From McLachlan, SIAM J. Sci. Comp. 16, 151 (1995):
  • M2 (order 2, all purpose)
  • M4 (order 4, all purpose)
• From Omelyan, Mryglod, Folk, Comput. Phys. Commun. 146, 188 (2002):
  • EFRL (order 4) optimized for H = A + B
  • PEFRL and VEFRL (order 4) optimized for H = A(p) + B(q). For PEFRL, chi should be exp(h X_A)exp(h X_B). For VEFRL, chi should be exp(h X_B)exp(h X_A).
• From Blanes, Moan, J. Comput. Appl. Math. 142, 313 (2002):
  • BM4 (order 4, all purpose) refers to S₆
  • BM6 (order 6, all purpose) refers to S₁₀
  • RKN4b (order 4) refers to SRKN₆^b optimized for H = A(p) + B(q). Here chi should be exp(h X_B)exp(h X_A).
  • RKN6b (order 6) refers to SRKN₁₁^b optimized for H = A(p) + B(q). Here chi should be exp(h X_B)exp(h X_A).
  • RKN6a (order 6) refers to SRKN₁₄^a optimized for H = A(p) + B(q). Here chi should be exp(h X_A)exp(h X_B).
• From Blanes, Casas, Farrés, Laskar, Makazaga, Murua, Appl. Numer. Math. 68, 58 (2013):
  • ABA104 (order (10,4)) optimized for H = A + ε B. Here chi should be exp(h X_A)exp(h X_B).
  • ABA864 (order (8,6,4)) optimized for H = A + ε B. Here chi should be exp(h X_A)exp(h X_B).
  • ABA1064 (order (10,6,4)) optimized for H = A + ε B. Here chi should be exp(h X_A)exp(h X_B).

All purpose integrators are for any splitting of the Hamiltonian H=∑_k A_k in any order of the functions A_k. Otherwise, the order of the operators is specified for each integrator.

Usage: integrator = SymplecticIntegrator(name, step) where name is one of the names listed above and step is the time step of the integrator (float).

The function integrator._integrate integrates the Hamiltonian flow by one time step.

The function integrator.integrate integrates the Hamiltonian flow from the initial conditions specified by the initial state vector y using integrator, one of the selected symplectic splitting integrators. It returns the value of y at times defines by the float, list or numpy array times.

Parameters:

• chi : function of (h, y), y being the state vector. Function returning exp(h X_n)...exp(h X₁) y. If the selected integrator is not all purpose, refer to the list above for the specific ordering of the operators. The operator X_k is the Liouville operator associated with the function A_k, i.e., for Hamiltonian flows X_k = {A_k , ·} where {· , ·} is the Poisson bracket.
• chi_star : function of (h, y). Function returning exp(h X₁)...exp(h X_n) y.
• y : initial state vector (numpy array)
• times : times at which the values of the state vector are computed
• command : function of (t, y). Function to be run at each time step (e.g., plotting an observable associated with the state vector, or register specific events).

Remarks:
- If the vector field is explicitly time dependent, it should be first autonomized by adding time as an extra variable
- If times is a linearly spaced list or array, or if times is an integer or a float, the time step is slightly readjusted so that the output times contain the values in times
- If times is not linearly spaced, a linear interpolation of the solution is performed; the accuracy of the integrator might be lost

Returns:
  Bunch object with the following fields defined:

• t : final integration time if times is a float or integer
    times if times is a list or a numpy array
    all computed times if times is a list or numpy array with a single element
• y : state vector at times t
• time_step : time step used in the computation

References:

• Hairer, Lubich, Wanner, 2003, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer)
• McLachlan, Tuning symplectic integrators is easy and worthwhile, Commun. Comput. Phys. 31, 987 (2022); arxiv:2104.10269