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I'm familiar with explicit numerical methods for solving ODE including Euler's method, and even Runge-Kutta methods (2nd and 4th order).

But I'm really confused when it comes to implicit methods. I understood the ''simplest'' implicit method, being Euler's backward method. I even did a code with it on Matlab and it worked just fine.

But, what about implicit Runge-Kutta 2nd order? How can I get the K's that would make it implicit? I've been searching for examples and can't find anything. It would really help if I saw a solution using an implicit RK-2 for a ODE. Is there any book/paper or anything?

Let's say we have this simple ODE: Y' = -2Y How would an implicit RK-2 work here?

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  • $\begingroup$ Did you find an answer to this? Even I am searching for implementation of RK2 for implicit time integration. The only thing I can think of is to calculate k1 using current value and k2 using the 'future-value', so because of k2, we get a set of equations that we have to solve. $\endgroup$
    – GRANZER
    Commented Jan 25, 2020 at 13:31

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The one-stage midpoint method is implicit and second order: $$\begin{eqnarray} k_1 &=& f\left(x_{i-1} + \frac{h}{2}, y_{i-1} + \frac{h}{2} k_1\right),\\ y_i &=& y_{i-1} + h k_1. \end{eqnarray}$$ It attains the maximum order 2 for a one-stage Runge-Kutta method, and it is A-stable but not L-stable.

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