Order of consistency for non autonomization invariant Runge-Kutta-Method

Question

I'm asked to find the stability function and interval, and prove that the order of consistency is at least 2, of this butcher tableau:

\begin{array} {c|cccc} 1\\ -1 & 2\\ \hline & \frac{3}{4} &\frac{1}{4} \end{array}

The stability function and interval are easy to find, but the system is non autonomization invariant. This makes proving that the order of consistency is at least 2 somewhat troubling.

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For the stability function:

$$k_1=f(t_j + h, y_j) = \lambda y_j$$ $$k_1=f(t_j -h, y_j + 2k_1) = \lambda (y_j + 2hk_1) = \lambda (y_j + 2h\lambda y_j)$$

$$y_{j+1} = y_j + h(\frac{3}{4}k_1 + \frac{1}{4}k_2) = y_j(1 + h \lambda + \frac{1}{2}(h\lambda)^2 ) $$ Thus the stability function is $g(z) = 1 + z + \frac{1}{2}z^2$

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For the stability intervall: $|g(z)| < 1$ or $z(1 + \frac{1}{2}z) < 0$ thus $-2 < z < 0$

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Now concerning the order of consistency. What was used in Find order of consistency of IVP with Butcher Tableau , Find order of consistency, stability function and stability interval of Runge-Kutta method and Find $\alpha$ such that $y_{j+1}=y_j+\frac{h}{2 \alpha}k_1 + h(1- \frac{1}{2 \alpha})k_2$ has order of consistency 2 was that the system was autonomization invariant.

That is, the left side of the upper part of the butcher tableau is equal to the sum of the elements on right side, which is not the case here ($1 \neq 0$ , $-1 \neq 2$). If a system is autonomization invariant, we simply need to derivate with respect to time. But here, that is not the case. Do I thus need to derivate with respect to both $t$ and $y$ ? Doesn't that make the whole things a lot more complicated ? Isn't there a nicer way to proceed ?

Any hint/ help would be greatly appreciated.

Answer 1

After coming across this question How to derive the order of a Runge-Kutta method from its Butcher tableau? , there seems to be a slightly easier way to find out the order of a RK method.

According to what's explained there, to find the order of this RK Method, we can simply multiply the terms on the lower left side of the table with their counterpart in the upper right part of the table, and then sum them all together. If the result is $1$, we have order $1$. If the result is $\frac{1}{2}$, we have order $2$. If the result is $\frac{1}{3}$, we have order $3$. In our case, we get

$$\frac{3}{4}\cdot 1 + \frac{1}{4} \cdot (-1) = \frac{1}{2}$$ which shows that we have order $2$.

Thats much easier, quicker and efficient than what was done for Find order of consistency of IVP with Butcher Tableau , Find order of consistency, stability function and stability interval of Runge-Kutta method and Find $\alpha$ such that $y_{j+1}=y_j+\frac{h}{2 \alpha}k_1 + h(1- \frac{1}{2 \alpha})k_2$ has order of consistency 2 . That's in fact awesome !