RKN1210 - A 12th/10th order Runge-Kutta-Nyström integrator

Integrator for second-order ODE's with very stringent error tolerances.
Updated 10 Jun 2018

RKN1210 12th/10th order Runge-Kutta-Nyström integrator
RKN1210() is a 12th/10th order variable-step numerical integrator for second-order ordinary differential equations of the form
y'' = f(t, y) (1)
with initial conditions

y(t0) = y0
y'(t0) = yp0 (2)

This second-order differential equation is integrated with a Runge-Kutta-Nyström method using 17 function evaluations per step. RKN12(10) is a very high-order method, to be used in problems with *extremely* stringent error tolerances.

The RKN-class of integrators is especially suited for problems of type (1). Compared to a classic Runge-Kutta integration scheme, the same accuracy can be obtained with fewer function evaluations. Also, it has been shown in various studies that this particular integration method is overall more efficient than (symplectic) multi-step or extrapolation methods that give the same accuracy.

RKN1210's behavior is very similar MATLAB's ODE-integrator suite; you can set options via ODESET, and input and output values are also practically the same.

Both output functions and event functions are fully supported.

The construction of RKN12(10) is described in
High-Order Embedded Runge-Kutta-Nyström Formulae
IMA Journal of Numerical Analysis (1987) 7, 423-430

Coefficients obtained from
These are also available in any format on request to these authors.

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Cite As

Rody Oldenhuis (2024). RKN1210 - A 12th/10th order Runge-Kutta-Nyström integrator (https://github.com/rodyo/FEX-RKN1210), GitHub. Retrieved .

MATLAB Release Compatibility
Created with R2010a
Compatible with any release
Platform Compatibility
Windows macOS Linux

Inspired by: rkn86

Inspired: Vectorized N-Body Equation

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Version Published Release Notes

Description update
- implemented NormControl
- the demo was broken
- error estimation used max() instead of min()
- renamed output.h -> output.stepsize and output.delta -> output.estimated_error

See GitHub for way more details (255 char limit here)

fixed divide-by-zero error for isolated cases where f turns all-zero

- Misc. bugfixes (minor)
- Implemented support for deval(). Note that MATLAB built-in deval() does not support rkn1210; a customized version is in the making.

- Found & removed some stray debugging code
- Corrected name & doc (ö instead of o)

- Fixed 2 bugs (thanks everyone!)
- Corrected & improved error handling
- It should be possible to use RKN1210 on R2008a and up
- Improved memory performance a tad
- Small stylistic updates to demo

Improvements regarding performance and memory usage.

bugfix for missing "index" variable on numel(tspan)>2 as noted by Tarek

- Removed bug inherited from RKN86; the error estimates there were all wrong. Now RKN1210 integrates at least 50x faster!
- Updated demo; included Earth+J2 example.

- Fixed 2 bugs: 1) exitflag = 3 was never assigned or described in the argument [output] 2) intermediate results were erased by event functions and only the event's zero was returned
- Improved performance a bit by pre-assigning the output arrays

- Interactive demonstration included
- implemented support for event functions
and output functions
- corrected various bugs, most notably the
(incorrectly) flipped output when tspan(2) < tspan(1)
- improved error handling

To view or report issues in this GitHub add-on, visit the GitHub Repository.
To view or report issues in this GitHub add-on, visit the GitHub Repository.