#59 Errors in the description of the 2N time-stepping scheme in the manual?

[Anonymous]

Originally created by: tobias.h...@gmail.com
Originally owned by: tobias.h...@gmail.com

(This does not affect the actual code but only the manual.)

Table 13 in section H.4 lists various coefficients for different 2N time-stepping schemes. In particular, the coefficients c_i specify how much time has elapsed after the i-th sub-time-step through the formula

t_i = t_0 + c_i dt

which is given in the table's caption. Since c_1 as listed in the table is always equal to 0, the above equation implies that t_1 = t_0, which seems odd. My guess is that c_1 should actually be equal to c_2 as listed in the table, c_2 should be equal to c_3, and c_3 should always be equal to 1.

Even then, however, it's unclear to me how one arrives at the coefficients listed for the cases "inhomogeneous" and "Williamson (1980)". For the first case I seem to get

t_1 - t_0 = 1/4
t_2 - t_1 = 5/12
t_3 - t_2 = 1/3

and for the second case

t_1 - t_0 = 1/3
t_2 - t_1 = 5/12
t_3 - t_2 = 1/4

[Anonymous]

Originally posted by: wdobler@gmail.com

I guess you are right. The c's should be numbered c_0, c_1, c_2 (and c_3=1 always, but this is irrelevant, while c_0 to c_2 determine the times where we evaluate the rhs). So the table should read

scheme                c0   c1    c2     a2      a3      b1   b2    b3
symmetric             0   1/3   2/3    -2/3     -1     1/3   1    1/2
[predictor/corrector] 0   1/2    1     -1/4    -4/3    1/2  2/3   1/2
inhomogeneous         0   1/4   2/3   -17/32  -32/27   1/4  8/9   3/4
Williamson (1980)     0   1/3   3/4   -5/9  -153/128  1/3 15/16 8/15

and indeed the c_i for the last two schemes were wrong.

Tobi, before fixidng the manual, could you check that this nomenclature (and the coefficients) are compatible with Williamson and/or Stanescu et al?

[Anonymous]

Originally posted by: tobias.h...@gmail.com

I'm not sure about Williamson (1980),  but what's in the manual *now* is entirely consistent with Stanescu & Habashi (1998). They also write

u_0 = u^{n-1}
u_3 = u^{n}

and

t_i = t^{n-1} + h c_i. In their tables, c_1=0 always, so that

t_1 = t^{n-1}
t_4 = t^{n}

This is pretty awkward I think, and IMO hardly worth maintaining compatibility with, also because the c_i are never hard coded anyway.

We could also avoid any compatibility issues altogether by listing the time intervals between successive sub-time-steps instead, which I would also find informative.