, 103 min read
Rubin's 4-th Order Method is Neither A-stable Nor D-stable
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This is in continuation of:
We analyze below method from Rubin, see his Fig. 4.2.
$$
\begin{array}{r|rr}
p=4 && 1 & 2\cr%R4A
\hline
-1 && 0 & 56\cr
0 && 24 & -72\cr
1 && -24 & 0\cr
2 && 0 & 16\cr
\hline
-1 && 1 & -21\cr
0 && -13 & -39\cr
1 && -13 & 33\cr
2 && 1 & 3\cr
\hline
c_{5i} && 0.0153 & 0.08125\cr
\end{array}
$$
The error constant is
$$
c_{p+1} = \frac{1}{\alpha_{i}\,(p+1)!} \sum_{i=0}^k\bigl(\alpha_ii^{p+1}-(p+1)\beta_ii^p\bigr).
$$
The stability polynomial has roots at 1 and 3.5, and therefore is not D-stable.
Rubin1, p=4, k=2, l=2
0.0000 56.0000
24.0000 -72.0000
-24.0000 0.0000
0.0000 16.0000
1.0000 -21.0000
-13.0000 -39.0000
-13.0000 33.0000
1.0000 3.0000
rho_0 0.000000000 0.000000000
rho_1 -0.000000000 0.000000000
rho_2 -0.000000000 0.000000000
rho_3 -0.000000000 0.000000000
rho_4 -0.000000000 0.000000000
rho_5 0.015277778 0.081250000 <-----
1. Stability region.
Below is the output of:
stabregion2 -f Rubin1 -oj -r600
Just looking at this stability region one could assume that the method is A-stable.
Truth is, it is not.
2. Stability mountain.
Below is the output of:
stabregion2 -f Rubin1 -o3 -r600 -L29
Rubin1 stability mountain.