RKN1210 - A 12th/10th order Runge-Kutta-Nyström integrator
Updated 2 May 2020
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
J. R. DORMAND, M. E. A. EL-MIKKAWY, AND P. J. PRINCE
IMA Journal of Numerical Analysis (1987) 7, 423-430
Coefficients obtained from
These are also available in any format on request to these authors.
Rody Oldenhuis (2023). RKN1210 - A 12th/10th order Runge-Kutta-Nyström integrator (https://github.com/rodyo/FEX-RKN1210/releases/tag/v2.1), GitHub. Retrieved .
MATLAB Release Compatibility
Platform CompatibilityWindows macOS Linux
Inspired by: rkn86
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Versions that use the GitHub default branch cannot be downloaded
See release notes for this release on GitHub: https://github.com/rodyo/FEX-RKN1210/releases/tag/v2.1
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)
- Found & removed some stray debugging code
- Fixed 2 bugs (thanks everyone!)
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!
- 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
- Interactive demonstration included