I'm referring this question : Generalization of Explicit Runge Kutta method
Even if, i start from $p=1$ Euler Method, $p=2$ (Runge Kutta with order of 2), $p=3$ (Runge Kutta with order of 3)
And so on. Notice that for order of $2$ and $3$, we need an initial value for solving and finding the numbers of $k$ and then construct the formula.
What i'm asking fo this. I know, that the numbers of $k$ grow faster than the numbers of the order.
For example :
For $p=4$, then $k=4$ stages,
For $p=5$, then $k=6$ stages,
For $p=6$, then $k=7$ stages,
For $p=7$, then $k=9$ stages,
For $p=8$, then $k=11$ stages,
Even if i know this fact. I can't generalize it by my self. The auckland book in the comment section of the previous question doesn't help me enough for construct the higher order of Runge Kutta.
Please, help me. I knew the formula for runge kutta of order 1, 2, 3 and 4. Even if i'm start to $p=1$ it doesn't really help me so much.
Journals, books, doesn't really help me. They just give the formula, and don't tell anything about how to generalize it into higher order.
Thanks in advance.
No, there is no general formula for explicit Runge-Kutta methods.
No, there is not THE order $p$ method, for $p>1$ the order $p$ methods form a positive-dimensional algebraic variety. This was known and explored from the start in 1900 by Heun and Kutta.
To design a higher order method, you have to solve the polynomial system of order conditions, best after identifying redundant equations. What the relevant papers do is exactly that, adding further conditions in demanding embedding, possibly of several orders, for stiffness detection and step size control. In some cases the method error is attempted to be reduce by some general estimate of the influence of the coefficients of the next-order derivatives.
Any general construction, such as applying Richardson extrapolation to the explicit Euler method, will be highly inefficient in the number of function evaluations. In contrast to the related Romberg integration, the order is only reduced by one in each extrapolation step. Then to get order $p$, you need about $2^p$ function evaluations per step, which is rather inefficient for higher orders.
The cited question seems to also iteratively increase the approximation order of the integral in the Picard iteration, thus possibly reducing the number of function evaluations to about $p^2$, which is still much larger than the usual RK methods. What you get is $p$ steps of the naive fixed-point iteration for the implicit collocation method on these sample points, transforming it into an explicit method (just as the Heun method is an explicit variant of the implicit trapezoidal method). One also has to explore what influence the usual deficiency of Lagrange interpolation on equidistant grids, the Runge phenomen, has on higher order results.