Improving Rates of Convergence of Iterative Schemes for Implicit Runge‐Kutta Methods

Abstract

Various iterative schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. In one scheme, when applied to an s-stage Runge-Kutta method, each step of the iteration still requires s function evaluations but consists of r(> s) sub-steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, · · · , and superlinear convergence is obtained in the limit k → ∞ . Some results are obtained for Gauss methods and numerical results are given.

Some Efficient Implementation Schemes for Implicit Runge-Kutta Methods

“…Method 1 * is the same method implemented using the scheme (7) with λ = 0.191729022 and B given by (20) for the case ρ[M (z)] = 0 at z = 0. Method 1 * * is also the same method implemented using the scheme (7) with λ = 0.214323763 , B given by (22) for the case ρ[M (z)] = 0 at z = ∞.…”

“…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.…”