Fifth order EPIRK methods

[MSeeker1340]

Again, the integrators EPIRK5s3 and EXPRB53s3 comes from this paper. Unfortunately current EPIRK5s3 produces large test error and seems to have only first order convergence. This is most likely a result of a typo/bug and not because of the underlying framework (since the other fifth order method EXPRB53s3 works without problem).

[ChrisRackauckas]

For my own future reference, 216 = 8*27, the 8 = 1/2^3 is the factor from the Krylov subspace algorithm because this calculates phi_i / t^i

[ChrisRackauckas]

Note that this line isn't fusing because of the scalar operations BTW. Just @. it. But optimizations can come later.

[ChrisRackauckas]

I'm not sure there's a typo in your code. I double checked your coefficient calculations and it looked solid to me.

[MSeeker1340]

I wrote a classical ExpRK implementation of EPIRK5s3 (using single step phiv instead of phiv_timestep) here: https://github.com/MSeeker1340/OrdinaryDiffEq.jl/commit/cb14d529f8f18039d34331941c8fe453ede9a9a9. For the same test problem in the test script, the two versions produce about the same error (~10^-4) whereas other ExpRK methods, be it classical or EPIRK, gives about 10^-7 error.

I have high suspicion that there is a mistake in the original formula. Since the coefficients are originally derived by symbolically solving a set of order conditions using Mathematica by the authors, I can try to substitute back the coefficients to check consistency.

Another angle I can tackle is to implement more EPIRK methods, like the original EPIRK5-P1/P2 integrators by Tokman. If all these don't have problems, then the likelihood that EPIRK5s3 is wrong will certainly increase.

[ChrisRackauckas]

Please do EPIRK5-P1. That one seems to be the most efficient when it doesn't have order loss (i.e. not too stiff) so it would be a pretty essential algorithm to have. EPIRK5s3 is really beat out in any case by EXPRB53s3 anyways, so if it doesn't exist then it doesn't have much of a practical difference (though I do like the completionism to be able to verify benchmarks)

[ChrisRackauckas]

LGTM. Merge when tests pass, and depwarns can be fixed later.

[MSeeker1340]

Note on the choice coefficients for EPIRK5-P1/P2: the $p$ coefficients aren't really free as they are cancelled out in the final form of the formula. The only free coefficient is $g_{22}$ in both cases. I made the simplest choice of $g_{22} = g_{32}$, so that no intermediate output is needed for the second column. I'm not sure if this is the optimal choice, and the paper doesn't have a discussion about this either.

[coveralls]

Coverage increased (+0.5%) to 74.809% when pulling 173fefc on MSeeker1340:expRK into c41ad05 on JuliaDiffEq:master.