How to Handle Fourth Order Implicit Runge-Kutta Methods?

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SUMMARY

This discussion focuses on the implementation of fourth order implicit Runge-Kutta methods, specifically in the context of symplectic integrators. The user references the formula for these methods, highlighting the complexity of calculating the dependent variables k_i for s>1. The solution involves substituting the expressions into the system f and solving for k, although this can lead to complicated scenarios. The user also cites a specific resource for symplectic coefficients, indicating a need for numerical approximation in certain cases.

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  • Understanding of Runge-Kutta methods, particularly implicit variants.
  • Familiarity with symplectic integrators and their applications.
  • Knowledge of numerical methods for solving differential equations.
  • Experience with mathematical notation and expressions in computational contexts.
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  • Study the implementation of implicit Runge-Kutta methods in numerical software like MATLAB or Python's SciPy library.
  • Explore the derivation and application of symplectic integrators in Hamiltonian systems.
  • Investigate numerical approximation techniques for solving complex systems of equations.
  • Review the specific coefficients used in symplectic integrators as detailed in academic papers, such as the one linked in the discussion.
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Mathematicians, physicists, and engineers working on numerical simulations, particularly those dealing with dynamical systems and requiring advanced integration techniques.

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Hey folks I'm experimenting with symplectic integrators and I'm trying to figure out how to deal with fourth order implicit Runge-Kutta methods as shown here (symplectic version second link).

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

where the formula is given as...

[tex]y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i[/tex]

[tex]k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^s a_{ij} k_j\right)[/tex]

But I have no idea how to use this for s>1 as each [tex]k_i[/tex] is dependent on [tex]k_j[/tex]. Is it a case of substituting the expression into our system f then solving for k? Seem like it could get very messy for any mildly complicated system. Perhaps we just do a numerical approximation?

The exact formula I'm using (which includes coefficient neccesary for it to be symplectic) is found at the top of page 6 here.

http://math.iisc.ernet.in/~rangaraj/docs/pramana_symplectic.pdf

But the problem with calculating k (which is called Z) here remains...

Any help appreciated.
 
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This is now solved btw. Too late to edit OP.
 

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