High_Order_DIRK_Methods_Coeffs

This repository accompanies the manuscript "VERY HIGH-ORDER A-STABLE STIFFLY ACCURATE DIAGONALLY IMPLICIT RUNGE-KUTTA METHODS WITH ERROR ESTIMATORS" available here https://arxiv.org/abs/2211.14574

It contains the coefficients for a 6th, 7th, and 8th order DIRK-type methods and their embedded error estimators. The notation used is $$\text{TYPE}(s,p)[r]X-[(\widehat{s},\widehat{p})Y]$$ where

• TYPE: is the structure of the DIRK-type method, that is DIRK, EDIRK, SDIRK, or ESDIRK.

• $s$: number of stages for the advancing method.

• $p$: order of convergence for the advancing method.

• $r$: stage order of the advancing method.

• $X$: stability property of the advancing method; i.e., $A$ for A-stable, $L$ for L-stable, $SA$ for stiffly accurate, or $SAL$ if both stiffly accurate and L-stable.

• $\widehat{s}$: number of stages for the embedded error estimator.

• $\widehat{p}$: order of convergence for the embedded error estimator.

• $Y$: stability property of the embedded error estimator, if any.

Note that in some methods $$\widehat{s} = s + 1 > s$$ for which we may take $$b_{s+1} = 0$$ in the embedded pair. Using the above notation, the new schemes are:

• Sixth-order methods

  • $\texttt{DIRK(6,6)[1]A-[(7,5)A]}$
  • $\texttt{DIRK(8,6)[1]SAL-[(8,5)A]}$
  • $\texttt{ESDIRK(8,6)[2]SA-[(8,4)]}$
  • $\texttt{SDIRK(9,6)[1]SAL-[(9,5)A]}$

• Seventh-order methods

  • $\texttt{DIRK(9,7)[1]A-[(9,5)A]}$
  • $\texttt{DIRK(10,7)[1]SAL-[(10,5)A]}$
  • $\texttt{ESDIRK(10,7)[2]SA-[(10,5)]}$
  • $\texttt{SDIRK(11,7)[1]SAL-[(11,5)A]}$

• Eighth-order methods

  • $\texttt{DIRK(13,8)[1]A-[(14,6)A]}$
  • $\texttt{DIRK(15,8)[1]SAL-[(16,6)A]}$
  • $\texttt{ESDIRK(16,8)[2]SAL-[(16,5)]}$