, 19 min read
Stability Mountain for Tendler-Rubin's Formula
In his dissertation chapter 3.4, J.M. Tendler shows below block implicit method, which he attributes to W. Rubin:
$$
\begin{array}{l}
-12 y_{n-1} + 24 y_n - 12 y_{n+1} &= h\left( 5 \dot y_{n-1} + 3 \dot y_n -9 y_{n+1} + \dot y_{n+2} \right),\\
8 y_{n-1} - 12 y_n + 4 y_{n+2} &= h\left( -3 \dot y_{n-1} - 5 \dot y_n + 7 \dot y_{n+1} + \dot y_{n+2} \right).
\end{array}
$$
It is an example for a method which is not A-stable. But it could be mistakenly be judged as A-stable.
References.
- Joel Marvin Tendler: “A Stiffly Stable Integration Process Using Cyclic Composite Methods”, Ph.D. Diss., Syracuse University, Syracuse, New York, 26-Feb-1973
The program stabregion4 can print information regarding error constants and stability properties.
Error constants.
TendlerRubin4, p=4, k=2, l=2
-12.0000 8.0000
24.0000 -12.0000
-12.0000 0.0000
0.0000 4.0000
5.0000 -3.0000
3.0000 -5.0000
-9.0000 7.0000
1.0000 1.0000
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 0.041666667 0.041666667 <-----
The C source code for stabregion4 is here stabregion4.c.
The parasitic roots and roots at infinity.
parasitic roots of TendlerRubin4
nr real imag abs 2-th root
0 1.00000000 0.00000000 1.00000000 1.00000000
1 1.00000000 0.00000000 1.00000000 1.00000000
radius at infinity of TendlerRubin4
nr real imag abs 2-th root
0 1.00000000 -0.00000002 1.00000000 1.00000000
1 1.00000000 0.00000002 1.00000000 1.00000000
Stability region.
stabregion4 -f TendlerRubin4 -oj -r300 -L3:-1:5:4
Stability mountain.
stabregion4 -f TendlerRubin4 -o3 -r200 -L4:-8:8:8