# 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 Jan 20, 2016 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. Jan 20, 2016 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 :) Jan 20, 2016 at 14:36

## 1 Answer

I think the work of Kennedy and Carpenter (mentioned already by @GoHokies) is still the definitive study on this topic. The journal paper can be found here; for some reason Google Scholar only provides links to the technical report. It includes methods of up to fifth order that are optimized for a range of properties relevant to convection-diffusion-reaction problems. Some of the fourth-order methods are compared with fully-implicit methods in this follow-up work.

One more fourth-order -- and low-storage -- IMEX method can be found in this 2013 paper of Cavaglieri et. al..

One comment regarding the order conditions: while it is true that the number of them grows very rapidly for IMEX methods, this number is much smaller if a few simplifying assumptions are invoked -- for instance, if the abscissas are identical.

• Wonderful, that is what I was looking for! Thanks, David. Jan 21, 2016 at 9:34
• If someone else wants to use the method, you may copy and paste the coefficients from PETsC: mcs.anl.gov/petsc/petsc-3.2/src/ts/impls/arkimex/arkimex.c -- this avoids having to putting in all those endless integers by hand :) Jan 24, 2016 at 8:41