Implicit Runge-Kutta method

In summary, the conversation discusses the use of fourth order implicit Runge-Kutta methods for symplectic integration. The formula for these methods is given and it is noted that it can become complicated for systems with s>1. The solution is to use a numerical approximation, with a specific formula found at the top of page 6 in a given source. The problem of calculating k (also known as Z) is also mentioned.
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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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  • #2
This is now solved btw. Too late to edit OP.
 

1. What is an Implicit Runge-Kutta method?

The Implicit Runge-Kutta method is a numerical integration technique that is commonly used to solve ordinary differential equations (ODEs). It is an extension of the popular Runge-Kutta method, but it allows for greater stability and accuracy by using a larger number of stages.

2. How does the Implicit Runge-Kutta method differ from the Explicit Runge-Kutta method?

The main difference between the Implicit and Explicit Runge-Kutta methods is that the Implicit method uses a larger number of stages and computes the values of all stages simultaneously, while the Explicit method computes the stages sequentially. This allows the Implicit method to handle stiffer ODEs, which are equations that change rapidly over time.

3. What are the advantages of using the Implicit Runge-Kutta method?

The Implicit Runge-Kutta method has several advantages over other numerical integration techniques. It is more accurate and stable, allowing for larger time steps and better handling of stiff ODEs. It also has a higher order of convergence, meaning that it can achieve more accurate results with fewer stages compared to other methods.

4. What are the limitations of the Implicit Runge-Kutta method?

The main limitation of the Implicit Runge-Kutta method is that it requires solving a system of nonlinear equations at each time step, which can be computationally expensive. It also requires more memory compared to the Explicit method. Additionally, it may not be suitable for all types of ODEs, as certain equations may require a specific type of integration method.

5. In what scientific fields is the Implicit Runge-Kutta method commonly used?

The Implicit Runge-Kutta method has a wide range of applications in various scientific fields, including physics, engineering, and biology. It is commonly used in simulations of physical systems, such as celestial mechanics, fluid dynamics, and quantum mechanics. It is also used in mathematical models of biological systems, such as population dynamics and biochemical reactions.

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