Dynamical, Hamiltonian, and 2nd Order ODE Solvers

# Dynamical, Hamiltonian, and 2nd Order ODE Solvers

These algorithms require an ODE defined in the following ways:

DynamicalODEProblem{isinplace}(f1,f2,u0,v0,tspan;kwargs...)
SecondOrderODEProblem{isinplace}(f,du0,u0,tspan;kwargs...)
HamiltonianProblem{T}(H,q0,p0,tspan;kwargs...)

These correspond to partitioned equations of motion:

$\frac{dv}{dt} = f_1(t,u) \\ \frac{du}{dt} = f_2(v) \\$

The functions should be specified as f1(dv,v,u,p,t) and f2(du,v,u,p,t) (in the inplace form), where f1 is independent of v (unless specified by the solver), and f2 is independent of t and v. This includes discretizations arising from SecondOrderODEProblems where the velocity is not used in the acceleration function, and Hamiltonians where the potential is (or can be) time-dependent but the kinetic energy is only dependent on v.

Note that some methods assume that the integral of f2 is a quadratic form. That means that f2=v'*M*v, i.e. $\int f_2 = \frac{1}{2} m v^2$, giving du = v. This is equivalent to saying that the kinetic energy is related to $v^2$. The methods which require this assumption will lose accuracy if this assumption is violated. Methods listed below make note of this requirement with "Requires quadratic kinetic energy".

## Recommendations

When energy conservation is required, use a symplectic method. Otherwise the Runge-Kutta-Nyström methods will be more efficient. Energy is mostly conserved by Runge-Kutta-Nyström methods, but is not conserved for long time integrations. Thus it is suggested that for shorter integrations you use Runge-Kutta-Nyström methods as well.

As a go-to method for efficiency, DPRKN6 is a good choice. DPRKN12 is a good choice when high accuracy, like tol<1e-10 is necessary. However, DPRKN6 is the only Runge-Kutta-Nyström method with a higher order interpolant (all default to order 3 Hermite, whereas DPRKN6 is order 6th interpolant) and thus in cases where interpolation matters (ex: event handling) one should use DPRKN6. For very smooth problems with expensive acceleration function evaluations, IRKN4 can be a good choice as it minimizes the number of evaluations.

For symplectic methods, higher order algorithms are the most efficient when higher accuracy is needed, and when less accuracy is needed lower order methods do better. Optimized efficiency methods take more steps and thus have more force calculations for the same order, but have smaller error. Thus the "optimized efficiency" algorithms are recommended if your force calculation is not too sufficiency large, while the other methods are recommend when force calculations are really large (for example, like in MD simulations VelocityVerlet is very popular since it only requires one force calculation per timestep). A good go-to method would be McAte5, and a good high order choice is KahanLi8.

## Standard ODE Integrators

The standard ODE integrators will work on Dynamical ODE problems via a transformation to a first-order ODE. See the ODE solvers page for more details.

## Specialized OrdinaryDiffEq.jl Integrators

Unless otherwise specified, the OrdinaryDiffEq algorithms all come with a 3rd order Hermite polynomial interpolation. The algorithms denoted as having a "free" interpolation means that no extra steps are required for the interpolation. For the non-free higher order interpolating functions, the extra steps are computed lazily (i.e. not during the solve).

### Runge-Kutta-Nyström Integrators

• Nystrom4: 4th order explicit Runge-Kutta-Nyström method. Allows acceleration to depend on velocity. Fixed timestep only.

• IRKN3: 4th order explicit two-step Runge-Kutta-Nyström method. Fixed timestep only.

• IRKN4: 4th order explicit two-step Runge-Kutta-Nyström method. Can be more efficient for smooth problems. Fixed timestep only.

• ERKN4: 4th order Runge-Kutta-Nyström method which is integrates the periodic properties of the harmonic oscillator exactly. Gets extra efficiency on periodic problems.

• ERKN5: 5th order Runge-Kutta-Nyström method which is integrates the periodic properties of the harmonic oscillator exactly. Gets extra efficiency on periodic problems.

• Nystrom4VelocityIndependent: 4th order explicit Runge-Kutta-Nyström method. Fixed timestep only.

• Nystrom5VelocityIndependent: 5th order explicit Runge-Kutta-Nyström method. Fixed timestep only.

• DPRKN6: 6th order explicit adaptive Runge-Kutta-Nyström method. Free 6th order interpolant.

• DPRKN8: 8th order explicit adaptive Runge-Kutta-Nyström method.

• DPRKN12: 12th order explicit adaptive Runge-Kutta-Nyström method.

### Symplectic Integrators

Note that all symplectic integrators are fixed timestep only.

• SymplecticEuler: First order explicit symplectic integrator

• VelocityVerlet: 2nd order explicit symplectic integrator.

• VerletLeapfrog: 2nd order explicit symplectic integrator.

• PseudoVerletLeapfrog: 2nd order explicit symplectic integrator.

• McAte2: Optimized efficiency 2nd order explicit symplectic integrator.

• Ruth3: 3rd order explicit symplectic integrator.

• McAte3: Optimized efficiency 3rd order explicit symplectic integrator.

• CandyRoz4: 4th order explicit symplectic integrator.

• McAte4: 4th order explicit symplectic integrator. Requires quadratic kinetic energy.

• CalvoSanz4: Optimized efficiency 4th order explicit symplectic integrator.

• McAte42: 4th order explicit symplectic integrator. (Broken)

• McAte5: Optimized efficiency 5th order explicit symplectic integrator. Requires quadratic kinetic energy

• Yoshida6: 6th order explicit symplectic integrator.

• KahanLi6: Optimized efficiency 6th order explicit symplectic integrator.

• McAte8: 8th order explicit symplectic integrator.

• KahanLi8: Optimized efficiency 8th order explicit symplectic integrator.

• SofSpa10: 10th order explicit symplectic integrator.

### GeometricIntegrators.jl

GeometricIntegrators.jl is a set of fixed timestep algorithms written in Julia. Note that this setup is not automatically included with DifferentialEquaitons.jl. To use the following algorithms, you must install and use GeometricIntegratorsDiffEq.jl:

Pkg.clone("https://github.com/JuliaDiffEq/GeometricIntegratorsDiffEq.jl")
using GeometricIntegratorsDiffEq
• GISymplecticEulerA - First order explicit symplectic Euler A

• GISymplecticEulerB - First order explicit symplectic Euler B

• GILobattoIIIAIIIB2 - Second order Gauss-Labatto-IIIA-IIIB

• GILobattoIIIBIIIA2 - Second order Gauss-Labatto-IIIB-IIIA