Some 2nd-order DWRK methods with arbitrarily large SSP coefficient are reported in my paper here: http://epubs.siam.org/doi/abs/10.1137/100818674. It would be nice to find downwind methods with large SSP coefficients that are

• Diagonally implicit

• Higher than 2nd order

Some code to look for new DWRK methods is in RK-Opt/dwrk-opt/. An essential point is that an upper bound for the individual coefficients must be given; here we denote it by \(M\).

Diagonally implicit methods

Numerical searches indicate that optimal DIRK methods with downwinding are just the optimal DIRK methods without downwinding – i.e., no advantage is gained by incorporating downwind terms and the 2S barrier cannot be avoided.

More searches indicate that there exist some DWDIRK methods with very slightly larger SSP coefficients.

3rd order methods

s=2

Allowing only 2 stages, the numerically optimal SSP coefficient seems to be bounded. It very gradually increases as \(M\to\infty\), and apparently \(\lim_{M\to\infty} {\mathcal C}_{2,3} = \sqrt{12}\).

s=3

It seems that \(\lim_{M\to\infty} {\mathcal C}_{3,3} = \infty\). Indeed, typically \({\mathcal C} \approx 2M\). We have found a nice family of methods. Although exact expressions for the coefficients are not known, they have the following asymptotic behavior:

\[\begin{align*} y_1 & = r^{-2} d_1 u^{n-1} + r^{-2} \alpha_{11}\left(y_1 + \frac{h}{r}f(y_1)\right) + r^{-2} \alpha_{12}\left(y_2 + \frac{h}{r}f(y_2)\right) + \tilde{\alpha}_{13}\left(y_3 + \frac{h}{r}\tilde{f}(y_3)\right) \\ y_2 & = \alpha_{22}\left(y_2 + \frac{h}{r}f(y_2)\right) + r^{-1} \alpha_{21}\left(y_1 + \frac{h}{r}f(y_1)\right) \\ y_3 & = y_1 + \frac{h}{r}f(y_1) \\ u^{n} & = y_2 + \frac{h}{r}f(y_2)\end{align*}\]

Here we have extracted the asymptotic behavior, so that all the coefficients shown are \(\mathcal O(1)\). Interestingly, the system to be solved becomes almost decoupled from the previous solution – in the limit \(r\to\infty\), the method would not be zero-well-defined. For finite but large \(r\), this means the system to be solved in the stage equations is very ill-conditioned. Specifically, the condition number of \(I-(\alpha+\tilde{\alpha})\) blows up. My 2nd-order methods have the same deficiency. But they may work well for some intermediate values of \(r\).

\(s>3\)

The \(s=4\) case seems to be very similar to the \(s=3\) case, except that \({\mathcal C}_{4,3}\approx 6M\). For \(s=5\), we see \({\mathcal C}_{5,3}\approx 8M\). An obvious conjecture is that \({\mathcal C}_{s,3}\approx 2(s-1)M\). This would mean there is only a slight improvement in the effective SSP coefficient for methods with more stages.

4th order methods

\(s=4\)

Seems to be bounded as \(M\to\infty\), with the optimal value around 5.8.

\(s>4\)

Seems unbounded as \(M\to\infty\), with coefficients blowing up in the limit.