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Implicit Runge–Kutta in R^3 space

  1. Nov 18, 2013 #1
    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.
     
  2. jcsd
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