Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s

To be A-stable, and possibly useful for stiff systems, a Runge–Kutta formula must be implicit. There is a significant computational advantage in diagonally implicit formulae, whose coefficient matrix is lower triangular with all diagonal elements equal. We derive new, strongly S-stable diagonally implicit Runge–Kutta formulae of order 2 in 2 stages and of order 3 in 3 stages, and show that it is impossible for a strongly S-stable diagonally implicit method to attain order 4 in 4 stages. Merely A-stable diagonally implicit formulae, of order 3 in 2 stages and of order 4 in 3 stages, were previously known; we prove that no 4-stage method of this type has order 5. We describe a computer program for stiff differential equations which uses these methods, and compare them to each other and to the GEAR package.