OrdinaryDiffEqSymplecticRK

Symplectic integrators are specialized methods for solving Hamiltonian systems and second-order differential equations that preserve important geometric properties of the phase space. These methods are essential for long-time integration of conservative mechanical systems.

Key Properties

Symplectic integrators provide:

Exact conservation of symplectic structure in phase space
Bounded energy error over long time periods
Excellent long-time stability without secular drift
Preservation of periodic orbits and other geometric structures
Linear energy drift instead of quadratic (much better than standard methods)

When to Use Symplectic Methods

Symplectic integrators are essential for:

Hamiltonian systems and conservative mechanical problems
Molecular dynamics and N-body simulations
Celestial mechanics and orbital computations
Plasma physics and charged particle dynamics
Long-time integration where energy conservation is critical
Oscillatory problems requiring preservation of periodic structure
Classical mechanics problems with known analytical properties

Mathematical Background

For a Hamiltonian system with energy H(p,q), symplectic integrators preserve the symplectic structure dp ∧ dq. While standard integrators have energy error growing quadratically over time, symplectic methods maintain bounded energy with only linear drift, making them superior for long-time integration.

Solver Selection Guide

First-order methods

SymplecticEuler: First-order, simplest symplectic method. Only recommended when the dynamics function f is not differentiable.

Second-order methods

McAte2: Optimized second-order McLachlan-Atela method, recommended for most applications
VelocityVerlet: Second-order, common choice for molecular dynamics but less efficient in terms of accuracy than McAte2
VerletLeapfrog: Second-order, kick-drift-kick formulation
LeapfrogDriftKickDrift: Alternative second-order leapfrog
PseudoVerletLeapfrog: Modified Verlet scheme

Third-order methods

Ruth3: Third-order method
McAte3: Optimized third-order McLachlan-Atela method

Fourth-order methods

CandyRoz4: Fourth-order method
McAte4: Fourth-order McLachlan-Atela (requires quadratic kinetic energy)
CalvoSanz4: Optimized fourth-order method
McAte42: Alternative fourth-order method (BROKEN)

Higher-order methods

McAte5: Fifth-order McLachlan-Atela method
Yoshida6: Sixth-order method
KahanLi6: Optimized sixth-order method
McAte8: Eighth-order McLachlan-Atela method
KahanLi8: Optimized eighth-order method
SofSpa10: Tenth-order method for highest precision

Method Selection Guidelines

For most applications: McAte2 (second-order, optimal efficiency)
For molecular dynamics (common choice): VelocityVerlet (less efficient than McAte2 but widely used)
For non-differentiable dynamics: SymplecticEuler (first-order, only when necessary)
For computational efficiency: McAte2 or McAte3

Important Note on Chaotic Systems

Most N-body problems (molecular dynamics, astrophysics) are chaotic systems where solutions diverge onto shadow trajectories. In such cases, higher-order methods provide no practical advantage because the true error remains O(1) for sufficiently long integrations - exactly the scenarios where symplectic methods are most needed. The geometric properties preserved by symplectic integrators are more important than high-order accuracy for chaotic systems.

For more information on chaos and accuracy in numerical integration, see: How Chaotic is Chaos? How Some AI for Science (SciML) Papers are Overstating Accuracy Claims

Installation

To be able to access the solvers in OrdinaryDiffEqSymplecticRK, you must first install them use the Julia package manager:

using Pkg
Pkg.add("OrdinaryDiffEqSymplecticRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqSymplecticRK
function HH_acceleration!(dv, v, u, p, t)
    x, y = u
    dx, dy = dv
    dv[1] = -x - 2x * y
    dv[2] = y^2 - y - x^2
end
initial_positions = [0.0, 0.1]
initial_velocities = [0.5, 0.0]
tspan = (0.0, 1.0)
prob = SecondOrderODEProblem(HH_acceleration!, initial_velocities, initial_positions, tspan)
sol = solve(prob, KahanLi8(), dt = 1 / 10)

Full list of solvers

OrdinaryDiffEqSymplecticRK.SymplecticEuler — Type

SymplecticEuler()

Symplectic Runge-Kutta Methods First order explicit symplectic integrator.

Keyword Arguments

References

https://en.wikipedia.org/wiki/Semi-implicitEulermethod