Jed Brown has suggested that a useful way to get parallelism in time would be to use the Butcher/Bickart technique of diagonalizing the \(A\) matrix and then doing all the block solves in parallel. But this requires RK methods with many stages for which \(A\) can be diagonalized in a numerically stable way. Hence the eigenvector matrix for \(A\) must be well-conditioned. This doesn’t hold for the Gauss methods – they get badly conditioned as the order increases.

Symmetric \(A\)

The most obvious approach is to make \(A\) symmetric – hence unitarily diagonalizable. Numerical optimization suggests that symmetry of \(A\) may be incompatible with 4th order accuracy (though 3rd order is easily achievable). I’m trying to prove this.

So far I have shown that if \(A\) is symmetric then the method can’t have stage order two, since

\[\sum_i c_i^2 = \sum_i (Ac)_i\] hence \[2\sum_i(\tau_2)_i = - \sum_i c_i^2 \ne 0 .\]

Update: using Mathematica, Lajos has found methods of order 4 with symmetric A.

Normal \(A\)

If \(A\) is normal, it seems again that \(\tau_2 = 0\) implies \(c=0\), but the proof above doesn’t work.