Some schemes for the implementation of implicit Runge-Kutta methods

Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

“…Cooper and Vignesvaran [9]obtained the optimum values λ = 0.202740067 and β = 1.159572736 when solving the problem (10). For these values of λ and β, ρ[M (z)] < 0.1599 for all z ∈ C − .…”

“…Cooper and Vignesvaran [9] showed that these equations can be solved only for one positive value of λ, λ = 0.146840443 and they obtained the optimum value β = 1.034 to solve the problem (10). In this case, ρ[M (z)] < 0.3467 for Re(z) ≤ 0.…”

“…Therefore, the other parameter λ has to be chosen to solve the problem (10). It follows from (11) that the polynomial p is given by…”

“…In other schemes advantage is taken of the special forms of some implicit methods [2], [4], [5], [12]. In another approach, schemes based directly on iterative procedure have been developed [3], [8], [9], [10], [13], [21]. For a singly implicit method, there is a non-singular matrix S so that S −1 AS = λ(I s − L) −1 , where L is zero except for some ones on the sub-diagonal.…”

“…Only one vector transformation is needed for each full step so that this scheme is more efficient. Another scheme was proposed by Cooper and Vignesvaran [10] in order to obtain improved rate of convergence, by adding extra sub-steps.Vigneswaran [20] obtained further improvement in the rate of convergence of the iteration scheme proposed in [10]. Gonzalez, Gonzalez and Montijano [16] proposed a scheme for Gauss methods using an iterative procedure of semi-implicit type in which the Jacobian does not appear explicitly.…”