# Runge-Kutta-2 on System

After spending some time using the Mathematica documentation and this Mathematica.SE answer, I implemented the Runge-Kutta-2 routines.

I am hoping someone can validate what I did and tell me that it is correct (especially the Butcher Tableau I used) and the step size $h = 0.1$.

  ClassicalRungeKuttaCoefficients[4, prec_] :=
With[{amat = {{1/2}}, bvec = {0, 1}, cvec = {1/2}},
N[{amat, bvec, cvec}, prec]]

{xf, yf} = {x, y} /.
First@NDSolve[{x'[t] == -y[t], y'[t] == x[t], x == 1,
y == 0}, {x, y}, {t, 0, 6},
Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4,
"Coefficients" -> ClassicalRungeKuttaCoefficients},
StartingStepSize -> 1/10];



We can find a closed form solution to this problem as:

  s = DSolve[{x'[t] == -y[t], y'[t] == x[t], x == 1, y == 0}, {x, y}, t]


Lastly, is there an automated way to update and step through each variant of RK-2, RK-3, RK-4, ... without having to manually enter the Butcher values (in other words, I want to step through each variant of RK-n and compare the errors (a table of that would be great))?

There is a hint of this at Wolfram's Reference page.

Note: I am embarrassed to say that I did not totally understand ClassicalRungeKuttaCoefficients (other than the coefficients).

The problem with your original code is that the order argument you specified (4) is in fact not the same as the order of the midpoint method (2). Thus:

MidpointCoefficients[2, prec_] := N[{{{1/2}}, {0, 1}, {1/2}}, prec];


For comparison purposes, here's the Butcher table for Heun's method:

HeunCoefficients[2, prec_] := N[{{{1}}, {1/2, 1/2}, {1}}, prec];


We can now pass the Butcher table to NDSolve[] like so:

{xm, ym} = {x, y} /.
First @ NDSolve[{x'[t] == -y[t], y'[t] == x[t], x == 1, y == 0},
{x, y}, {t, 0, 6},
Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 2,
"Coefficients" -> MidpointCoefficients},
StartingStepSize -> 1/10];


and check that the results are as expected:

ListLinePlot[Transpose[{xm["ValuesOnGrid"], ym["ValuesOnGrid"]}],
AspectRatio -> Automatic] {Norm[xc["ValuesOnGrid"] - Cos[Range[0, 6, 1/10]], ∞],
Norm[yc["ValuesOnGrid"] - Sin[Range[0, 6, 1/10]], ∞]}
{0.00813941, 0.00938408}


Lastly, is there an automated way to update and step through each variant of RK-2, RK-3, RK-4, ... without having to manually enter the Butcher values ... ?

In principle, one could certainly use Mathematica to solve the underlying algebraic equations satisfied by the coefficients of an $n$-th order Runge-Kutta method. But, as noted in the venerable book of Hairer/Nørsett/Wanner, the so-called order conditions very quickly become more numerous and complicated as the order is increased, so one often relies on trickery and/or approximations to generate usable higher-order coefficients. So, you could try, but you'd find it way easier to stand on the shoulders of giants.