Butcher tableau for Verner's 16 stage, Order (9,8) Runge-Kutta method. More...

Variables
integer, parameter	s = 16
	The order of the overall method.
integer, parameter	m = 2
	Number of methods.
real(kind=rk), dimension(s, s), parameter	a
	The \(\mathbf{a}\) matrix for the Butcher Tableau.
real(kind=rk), dimension(s, m), parameter	b
	The \(\mathbf{b}\) matrix for the Butcher Tableau.
real(kind=rk), dimension(s), parameter	c
	The \(\mathbf{c}\) matrix for the Butcher Tableau.
integer, dimension(m), parameter	p = [9, 8]
	The method orders.
integer, dimension(m), parameter	se = [15, 16]
	Number of stages for each method.

Detailed Description

Butcher tableau for Verner's 16 stage, Order (9,8) Runge-Kutta method.

Known Aliases: 'Vern9' (OrdinaryDiffEq.jl), 'ARKODE_VERNER_16_8_9' (SUNDIALS).

IMO: This is my go-to O(9,8) method.

References:

Verner (1978) "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error"; SIAM J. Num. Anal. 15 (4); p772-790
Jim Verner's Refuge for Runge-Kutta Pairs
- Retrieved on 2025-07-16 from : https://www.sfu.ca/~jverner/
- On the above page, in a section annotated with "Added May, 2007", this tableau was labeled as A "more efficient" Runge–Kutta (9)8 Pair with Interpolants
- The two links with the data were:
  - https://www.sfu.ca/~jverner/RKV98.IIa.Efficient.000000349.081209.FLOAT6040OnWeb zotero://select/items/0_UEQBISIW
  - https://www.sfu.ca/~jverner/RKV98.IIa.Efficient.000000349.081210.CoeffsOnlyRADandFLOATS
- The first link above is titled: A 'most efficient" Runge–Kutta (16:9(8)) pair