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Stability Mountain for Runge-Kutta 4
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The classical Runge-Kutta method of order 4 with 4 stages.
$$
\begin{array}{c|cccc}
{1\over2} & {1\over2}\\
{1\over2} & 0 & {1\over2}\\
1 & 0 & 0 & 1\\
\hline
& {1\over6} & {1\over3} & {1\over3} & {1\over6}
\end{array}
$$
The output of stabregion3 -df RK4:
RK4, p=4, k=1, l=4
-2.0000 -2.0000 -1.0000 -6.0000
2.0000 0.0000 0.0000 0.0000
0.0000 2.0000 0.0000 0.0000
0.0000 0.0000 1.0000 0.0000
0.0000 0.0000 0.0000 6.0000
1.0000 0.0000 0.0000 1.0000
0.0000 1.0000 0.0000 2.0000
0.0000 0.0000 1.0000 2.0000
0.0000 0.0000 0.0000 1.0000
0.0000 0.0000 0.0000 0.0000
rho_0 0.000000000 0.000000000 0.000000000 0.000000000
rho_1 0.500000000 1.500000000 2.000000000 3.000000000 <-----
rho_2 0.500000000 1.500000000 2.500000000 6.500000000 <-----
rho_3 0.166666667 1.083333333 2.500000000 9.083333333 <-----
rho_4 0.041666667 0.583333333 2.041666667 9.416666667 <-----
rho_5 0.008333333 0.245833333 1.358333333 7.734722222 <-----
The consistency order of Runge-Kutta formulas cannot be computed using the usual methods for linear multistep methods. Therefore above error constants are not correct. For the details see Recursive Generation of Runge-Kutta Formulas.
The matrices in question are given below.
k=1, l=4, rest=3, n=4, nrest=4, nsq=16, colLen=10
RK4
A1
2.00000 0.00000 0.00000 0.00000
0.00000 2.00000 0.00000 0.00000
0.00000 0.00000 1.00000 0.00000
0.00000 0.00000 0.00000 6.00000
A0
0.00000 0.00000 0.00000 -2.00000
0.00000 0.00000 0.00000 -2.00000
0.00000 0.00000 0.00000 -1.00000
0.00000 0.00000 0.00000 -6.00000
B1
0.00000 0.00000 0.00000 0.00000
1.00000 0.00000 0.00000 0.00000
0.00000 1.00000 0.00000 0.00000
2.00000 2.00000 1.00000 0.00000
B0
0.00000 0.00000 0.00000 1.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 1.00000
Parasitic roots of RK4
nr real imag abs 4-th root
0 1.00000000 0.00000000 1.00000000 1.00000000
1 0.00000000 -0.00000000 0.00000000 0.00000000
2 0.00000000 -0.00000000 0.00000000 0.00000000
3 0.00000000 -0.00000000 0.00000000 0.00000000
1. Stability region.
Below is the output of:
stabregion3 -f RK4 -oj -r120
2. Stability mountain.
Below is the output of:
stabregion3 -f RK4 -o3 -L8:-2.1:0.7:2.8 -r180