In this article, a new family of Runge-Kutta methods of 8^th order for solving ordinary differential equations is discovered and depends on the parameters b_8 and a_10;5. For b8 = 49/180 and a10;5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a_10;5. We compare the stability regions according to the values of a_10;5 with respect to the stability region of Cooper-Verner.

The Runge-Kutta method of order 5 with 6 stages requires finding a matrix A, whose coefficients must satisfy a system of nonlinear polynomial equations. Butcher found a 5-parameter family of solutions, which displays different characteristics depending on whether b2= 0 or b2, 0. This paper presents an optimal method in the case b2= 0, which is significantly better than several popular methods of order 4.