Implicit Runge–Kutta in R^3 space

  • Thread starter Beduino
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Main Question or Discussion Point

I'm trying to solve a implicit runge kutta algorithm numerically in ℝ3 space as a integrator for orbital simulation.

http://en.wikipedia.org/wiki/Runge–Kutta_methods#Implicit_Runge.E2.80.93Kutta_methods

More specifically a 6th order Gauss–Legendre method
http://en.wikipedia.org/wiki/Gauss–Legendre_method

I have worked out the three K parameters needed in the method, which forms a system of non linear vector equations below.

[itex]\left\{\begin{matrix}
\overrightarrow{K_{1}}=-\delta t*\mu\frac{\overrightarrow{r}+
a_{11}\overrightarrow{K_{1}}+
a_{12}\overrightarrow{K_{2}}+
a_{13}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+
a_{11}\overrightarrow{K_{1}}+
a_{12}\overrightarrow{K_{2}}+
a_{13}\overrightarrow{K_{3}} \right |^{3}}\\
\overrightarrow{K_{2}}=-\delta t*\mu\frac{\overrightarrow{r}+
a_{21}\overrightarrow{K_{1}}+
a_{22}\overrightarrow{K_{2}}+
a_{23}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+
a_{21}\overrightarrow{K_{1}}+
a_{22}\overrightarrow{K_{2}}+
a_{23}\overrightarrow{K_{3}} \right |^{3}}\\
\overrightarrow{K_{3}}=-\delta t*\mu\frac{\overrightarrow{r}+
a_{31}\overrightarrow{K_{1}}+
a_{32}\overrightarrow{K_{2}}+
a_{33}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+
a_{31}\overrightarrow{K_{1}}+
a_{32}\overrightarrow{K_{2}}+
a_{33}\overrightarrow{K_{3}} \right |^{3}}
\end{matrix}\right.[/itex]

What's the most appropriated way to solve this system numerically, all parameters are given, except each K vector.
 

Answers and Replies

  • #2
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There is no other way than to solve it the hard way.
 

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