Implicit Runge–Kutta in R^3 space

In summary, the conversation discussed solving an implicit Runge-Kutta algorithm numerically in ℝ3 space as an integrator for orbital simulation. Specifically, the 6th order Gauss-Legendre method was mentioned, and the necessary K parameters were derived. The resulting system of non-linear vector equations must be solved using numerical methods, as all parameters are given except for each K vector.
  • #1
Beduino
6
2
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.
 
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  • #2
There is no other way than to solve it the hard way.
 

1. What is Implicit Runge-Kutta in R^3 space?

Implicit Runge-Kutta in R^3 space is a numerical method used for solving differential equations in three-dimensional space. It is an extension of the standard Runge-Kutta method, which is used for solving differential equations in one or two dimensions.

2. How does Implicit Runge-Kutta differ from other numerical methods?

Unlike other numerical methods, Implicit Runge-Kutta uses an implicit formula for calculating the slope of the solution function at each step. This means that the time step can be larger, resulting in faster convergence and more accurate results.

3. What are the advantages of using Implicit Runge-Kutta in R^3 space?

One major advantage of using Implicit Runge-Kutta in R^3 space is that it can handle stiff systems of differential equations, which are difficult for other numerical methods to solve. It also allows for larger time steps, reducing the computational time needed to obtain a solution.

4. Are there any limitations to using Implicit Runge-Kutta in R^3 space?

One limitation of Implicit Runge-Kutta in R^3 space is that it requires more computational power and time compared to other numerical methods. It also may not be suitable for non-stiff systems of equations, as the implicit formula can lead to inaccuracies in the solution.

5. In what real-world applications is Implicit Runge-Kutta in R^3 space commonly used?

Implicit Runge-Kutta in R^3 space is commonly used in engineering and scientific fields where differential equations are used to model physical systems, such as fluid dynamics, chemical reactions, and population dynamics. It is also used in computer simulations and mathematical modeling to study complex systems.

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