OrdinaryDiffEqRKN

Runge-Kutta-Nyström (RKN) methods for solving second-order differential equations of the form d²u/dt² = f(u, du/dt, t). These methods are specifically designed for second-order ODEs and can be more efficient than converting to first-order systems when the problem has this natural structure.

Key Properties

RKN methods provide:

  • Direct integration of second-order ODEs without conversion to first-order systems
  • Efficient handling of problems where velocity dependence is minimal or absent
  • Specialized variants for different types of second-order problems
  • High-order accuracy with methods up to 12th order available
  • Trigonometrically-fitted variants for oscillatory problems
  • Improved efficiency for second-order problems compared to first-order conversions

When to Use RKN Methods

These methods are recommended for:

  • Second-order differential equations with natural d²u/dt² = f(u, du/dt, t) structure
  • Classical mechanics problems (Newton's equations of motion)
  • Oscillatory second-order systems (springs, pendulums, wave equations)
  • Problems where velocity dependence is weak or absent
  • Celestial mechanics and orbital dynamics
  • Vibration analysis and structural dynamics
  • Wave propagation problems in their natural second-order form

Problem Types

Velocity-independent problems: d²u/dt² = f(u, t)

When the acceleration depends only on position (and possibly time):

  • More efficient specialized methods available
  • Classical examples: gravitational problems, springs with F = -kx

Velocity-dependent problems: d²u/dt² = f(u, du/dt, t)

When acceleration depends on both position and velocity:

  • General RKN methods handle the full dependence
  • Examples: damped oscillators, air resistance problems

Solver Selection Guide

General-purpose RKN methods

  • Nystrom4: Fourth-order method for general second-order ODEs
  • IRKN3, IRKN4: Improved Runge-Kutta-Nyström methods

Velocity-independent specialized methods

  • Nystrom4VelocityIndependent: Fourth-order for d²u/dt² = f(u, t)
  • Nystrom5VelocityIndependent: Fifth-order for d²u/dt² = f(u, t)

Velocity-dependent methods

  • FineRKN4: Fourth-order allowing velocity dependence
  • FineRKN5: Fifth-order allowing velocity dependence

High-order Dormand-Prince variants

  • DPRKN4: Fourth-order Dormand-Prince RKN
  • DPRKN5: Fifth-order Dormand-Prince RKN
  • DPRKN6: Sixth-order with free interpolant
  • DPRKN6FM: Sixth-order with optimized error coefficients
  • DPRKN8: Eighth-order for high accuracy
  • DPRKN12: Twelfth-order for extreme precision

Trigonometrically-fitted methods for oscillatory problems

  • ERKN4: Embedded 4(3) pair for periodic problems
  • ERKN5: Embedded 5(4) pair for periodic problems
  • ERKN7: Higher-order embedded pair for oscillatory systems

Specialized methods

  • RKN4: Fourth-order for linear inhomogeneous second-order IVPs

Advantages Over First-Order Conversion

  • Natural problem structure: Preserves the physical meaning of the equations
  • Specialized optimizations: Methods can exploit second-order structure

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

using Pkg
Pkg.add("OrdinaryDiffEqRKN")

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 OrdinaryDiffEqOrdinaryDiffEqRKN
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, Nystrom4(), dt = 1 / 10)

Full list of solvers

OrdinaryDiffEqRKN.IRKN3 — Type
IRKN3()

Improved Runge-Kutta-Nyström method Method of order three, which minimizes the amount of evaluated functions in each step. Fixed time steps only. Second order ODE should not depend on the first derivative.

Keyword Arguments

References

@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer}}

OrdinaryDiffEqRKN.IRKN4 — Type
IRKN4()

Improved Runge-Kutta-Nyström method Improves Runge-Kutta-Nyström method of order four, which minimizes the amount of evaluated functions in each step. Fixed time steps only. Second order ODE should not be dependent on the first derivative. Recommended for smooth problems with expensive functions to evaluate.

Keyword Arguments

References

@article{rabiei2012numerical, title={Numerical Solution of Second-Order Ordinary Differential Equations by Improved Runge-Kutta Nystrom Method}, author={Rabiei, Faranak and Ismail, Fudziah and Norazak, S and Emadi, Saeid}, publisher={Citeseer}}

OrdinaryDiffEqRKN.Nystrom4 — Type
Nystrom4()

Improved Runge-Kutta-Nyström method 4th order explicit Runge-Kutta-Nyström method. Allows acceleration to depend on velocity.

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

OrdinaryDiffEqRKN.Nystrom4VelocityIndependent — Type
Nystrom4VelocityIndependent()

Improved Runge-Kutta-Nyström method A 4th order explicit method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

OrdinaryDiffEqRKN.Nystrom5VelocityIndependent — Type
Nystrom5VelocityIndependent()

Improved Runge-Kutta-Nyström method A 5th order explicit method. Used directly on second order ODEs, where the acceleration is independent from velocity (ODE Problem is not dependent on the first derivative).

Keyword Arguments

References

E. Hairer, S.P. Norsett, G. Wanner, (1993) Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edition. Springer Series in Computational Mathematics, Springer-Verlag.

OrdinaryDiffEqRKN.FineRKN4 — Type
FineRKN4()

Improved Runge-Kutta-Nyström method A 4th order explicit method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.

Keyword Arguments

References

@article{fine1987low, title={Low order practical {R}unge-{K}utta-{N}ystr{"o}m methods}, author={Fine, Jerry Michael}, journal={Computing}, volume={38}, number={4}, pages={281–297}, year={1987}, publisher={Springer}}

OrdinaryDiffEqRKN.FineRKN5 — Type
FineRKN5()

Improved Runge-Kutta-Nyström method A 5th order explicit method which can be applied directly to second order ODEs. In particular, this method allows the acceleration equation to depend on the velocity.

Keyword Arguments

References

@article{fine1987low, title={Low order practical {R}unge-{K}utta-{N}ystr{"o}m methods}, author={Fine, Jerry Michael}, journal={Computing}, volume={38}, number={4}, pages={281–297}, year={1987}, publisher={Springer}}

OrdinaryDiffEqRKN.DPRKN4 — Type
DPRKN4()

Improved Runge-Kutta-Nyström method 4th order explicit method. The second order ODE should not depend on the first derivative.

Keyword Arguments

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}

OrdinaryDiffEqRKN.DPRKN5 — Type
DPRKN5()

Improved Runge-Kutta-Nyström method 5th order explicit method. The second order ODE should not depend on the first derivative.

Keyword Arguments

References

@article{Bettis1973ARN, title={A Runge-Kutta Nystrom algorithm}, author={Dale G. Bettis}, journal={Celestial mechanics}, year={1973}, volume={8}, pages={229-233}, publisher={Springer}}

OrdinaryDiffEqRKN.DPRKN6 — Type
DPRKN6()

Improved Runge-Kutta-Nyström method 6th order explicit method. The second order ODE should not depend on the first derivative. Free 6th order interpolant

Keyword Arguments

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}

OrdinaryDiffEqRKN.DPRKN6FM — Type
DPRKN6FM()

Improved Runge-Kutta-Nyström method 6th order explicit method. The second order ODE should not depend on the first derivative. Compared to DPRKN6, this method has smaller truncation error coefficients which leads to performance gain when only the main solution points are considered.

Keyword Arguments

References

@article{Dormand1987FamiliesOR, title={Families of Runge-Kutta-Nystrom Formulae}, author={J. R. Dormand and Moawwad E. A. El-Mikkawy and P. J. Prince}, journal={Ima Journal of Numerical Analysis}, year={1987}, volume={7}, pages={235-250}}

OrdinaryDiffEqRKN.DPRKN8 — Type
DPRKN8()

Improved Runge-Kutta-Nyström method 8th order explicit method. The second order ODE should not depend on the first derivative. Not as efficient as DPRKN12 when high accuracy is needed, however this solver is competitive with DPRKN6 at lax tolerances and, depending on the problem, might be a good option between performance and accuracy.

Keyword Arguments

References

@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press}}

OrdinaryDiffEqRKN.DPRKN12 — Type
DPRKN12()

Improved Runge-Kutta-Nyström method 12th order explicit method. The second order ODE should not depend on the first derivative. Most efficient when high accuracy is needed.

Keyword Arguments

References

@article{dormand1987high, title={High-order embedded Runge-Kutta-Nystrom formulae}, author={Dormand, JR and El-Mikkawy, MEA and Prince, PJ}, journal={IMA Journal of Numerical Analysis}, volume={7}, number={4}, pages={423–430}, year={1987}, publisher={Oxford University Press}}

OrdinaryDiffEqRKN.ERKN4 — Type
ERKN4()

Improved Runge-Kutta-Nyström method Embedded 4(3) pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.

Keyword Arguments

References

@article{demba2017embedded, title={An Embedded 4 (3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nystr{"o}m Method for Solving Periodic Initial Value Problems}, author={Demba, MA and Senu, N and Ismail, F}, journal={Applied Mathematical Sciences}, volume={11}, number={17}, pages={819–838}, year={2017}}

OrdinaryDiffEqRKN.ERKN5 — Type
ERKN5()

Improved Runge-Kutta-Nyström method Embedded 5(4) pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.

Keyword Arguments

References

@article{demba20165, title={A 5 (4) Embedded Pair of Explicit Trigonometrically-Fitted Runge–Kutta–Nystr{"o}m Methods for the Numerical Solution of Oscillatory Initial Value Problems}, author={Demba, Musa A and Senu, Norazak and Ismail, Fudziah}, journal={Mathematical and Computational Applications}, volume={21}, number={4}, pages={46}, year={2016}, publisher={Multidisciplinary Digital Publishing Institute}}

OrdinaryDiffEqRKN.ERKN7 — Type
ERKN7()

Improved Runge-Kutta-Nyström method Embedded pair of explicit methods. Integrates the periodic properties of the harmonic oscillator exactly. The second order ODE should not depend on the first derivative. Uses adaptive step size control. This method is extra efficient on periodic problems.

Keyword Arguments

References

@article{SimosOnHO, title={On high order Runge-Kutta-Nystr{"o}m pairs}, author={Theodore E. Simos and Ch. Tsitouras}, journal={J. Comput. Appl. Math.}, volume={400}, pages={113753}}

OrdinaryDiffEqRKN.RKN4 — Type
RKN4()

Improved Runge-Kutta-Nyström method 3 stage fourth order method to solve second order linear inhomogeneous IVPs. Does not include an adaptive method. Solves for for d-dimensional differential systems of second order linear inhomogeneous equations.

Warning

This method is only fourth order for these systems, the method is second order otherwise!

Keyword Arguments

References

@article{MONTIJANO2024115533, title = {Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs}, author = {J.I. Montijano and L. Rández and M. Calvo}, journal = {Journal of Computational and Applied Mathematics}, volume = {438}, pages = {115533}, year = {2024},}