# Fourth order IMEX Runge-Kutta method

I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well as a number of works that cite this paper (through Google Scholar).

However, in all papers I looked at, only methods of order up to three were given. Since the number of order conditions increases very quickly as the order goes up (in his slides, Rascheri states that a general IMEX-RKM has to satisfy 56 conditions), I wonder if such a method has been derived anywhere?

Is there a paper somewhere that states the Butcher tableau for a fourth order IMEX Runge-Kutta method?

• Perhaps the ESDIRK4 scheme is an example of what you've been looking for? link – GoHokies Jan 20 '16 at 11:51
• ESDIRK4 is a DIRK method where the first stage is explicit (zero diagonal entries) and all other diagonal entries are identical. This is not an IMEX method though and therefore not what I am looking for. – Daniel Jan 20 '16 at 11:55
• Yes, some confusion there on my part. The ESDIRK scheme is actually "one-half" of the (additive) IMEX-RK procedure - namely, the one used to integrate the stiff terms. Non-stiff terms can be integrated with a "traditional" explicit (E) RK scheme. You can find a 4th order additive IMEX (ESDIRK + ERK) method in this report. Hope this proves more useful than my 1st comment :) – GoHokies Jan 20 '16 at 14:36