Some Efficient Schemes with Improving Rate of Convergence for Two-stage Gauss Method

  • Ratnam Vigneswaran

Abstract


Various iteration schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. A modified Newton scheme is typically used to solve these equations. As an alternative to this scheme, iteration schemes, which sacrifice super-linear convergence for
reduced linear algebra costs, have been proposed. A scheme of this type proposed in [11] avoids expensive vector transformations and is computationally more
efficient. The rate of convergence of this scheme is examined in [11] when it is applied to the scalar test differential equation 􀢞′ 􀵌 􀢗􀢞 and the convergence rate
depends on the spectral radius of the iteration matrix 􀡹􁈺􀢠􁈻, a function of 􀢠 􀵌 􀢎􀢗, where 􀢎 is the step-length. In this scheme, some conditions are imposed on the spectral radius of its iteration matrix in order to get super-linear convergence at two points in the 􀢠 -plane. Then the supremum of the spectral radius of each
scheme is minimized over the left-half 􀢠 -plane in order to improve the convergence rate of the scheme. Two new schemes with optimal parameters are obtained for the two-stage Gauss method and some numerical experiments are reported.



 

Published
2018-05-11
How to Cite
VIGNESWARAN, Ratnam. Some Efficient Schemes with Improving Rate of Convergence for Two-stage Gauss Method. GSTF Journal of Mathematics, Statistics and Operations Research (JMSOR), [S.l.], v. 2, n. 1, may 2018. ISSN 2251-3396. Available at: <http://dl6.globalstf.org/index.php/jmsor/article/view/1519>. Date accessed: 16 dec. 2018.