http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/135 Sun, 19 Apr 2026 11:22:04 GMT 2026-04-19T11:22:04Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3810 Title: Accelerating convergence rate of linear iteration schemes based on projection method for three-stage gauss method Authors: Kajanthan, S.; Vigneswaran, R. Abstract: The non-linear equations arising in the implementation of implicit Runge-Kutta methods have been solved by various iteration schemes. Several authors have been proposed various iteration schemes with reduced linear algebra costs. To accelerate the convergence rate of those linear iteration schemes, a class of s-step non-linear scheme based on projection method was proposed. In this scheme, sequence of numerical solutions is updated after each sub-step is completed. The efficiency of this scheme was examined when it is applied to the linear scalar problem with rapid convergence required for all in the left half complex plane, where is a step size, and obtained the iteration matrix of the new scheme. For three-stage Gauss method, upper bound for the spectral radius of its iteration matrix was obtained in the left half complex plane. Finally, some numerical experiments are carried out to confirm the obtained theoretical results. Results for some non-linear stiff problems whose Jacobian matrix has both small eigenvalues and eigenvalues with largest negative real part are reported and compared with results obtained. Numerical result shows that, the proposed class of non- linear iteration scheme accelerates the convergence rate of the linear iteration scheme that we consider for the comparison in this work. It will be possible to apply the proposed class of non-linear scheme to accelerate the rate of convergence of other linear iteration schemes. Sun, 01 Jan 2017 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3810 2017-01-01T00:00:00Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3809 Title: Improving rate of convergence of an iterative scheme with extra sub-steps for two stage gauss method. Authors: Vigneswaran, R.; Kajanthan, S. Abstract: The non-linear equations, when implementing implicit Runge-Kutta methods, may be solved by a modified Newton scheme and by several linear iteration schemes which sacrificed superlinear convergence for reduced linear algebra costs. A linear scheme of this type was proposed, which requires some additional computation in each iteration step. The rate of convergence of this scheme is examined when it is applied to the scalar test problem x ′ = qx and the convergence rate depends on the spectral radius ρ[M(z)] of the iteration matrix M, a function of z = hq, where h is a step size. The supremum of the lower bound for ρ[M(z)] is minimized over left-half plane of the z-complex plane and over the negative real axis of the z-plane in order to improve the rate of convergence of that scheme. Two new schemes are obtained for the two stage Gauss method and numerical results are given. Sun, 01 Jan 2017 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3809 2017-01-01T00:00:00Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3808 Title: Some efficient implementation schemes for implicit RUNGE-KUTTA methods Authors: Vigneswaran, R.; Kajanthan, S. Abstract: Several iteration schemes have been proposed to solve the nonlinear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to the modified Newton scheme, some iteration schemes with reduced linear algebra costs have been proposed A scheme of this type proposed in [9] avoids expensive vector transformations and is computationally more efficient. The rate of convergence of this scheme is examined in [9] when it is applied to the scalar test differential equation x ′ = qx and the convergence rate depends on the spectral radius of the iteration matrix M(z), a function of z = hq, where h is the step-length. In this scheme, we require the spectral radius of M(z) to be zero at z = 0 and at z = ∞ in the z-plane in order to improve the rate of convergence of the scheme. New schemes with parameters are obtained for three-stage and four-stage Gauss methods. Numerical experiments are carried out to confirm the results obtained here. Wed, 01 Jan 2014 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3808 2014-01-01T00:00:00Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3807 Title: A class of s-step non-linear iteration scheme based on projection method for gauss method. Authors: Vigneswaran, R.; Kajanthan, S. Abstract: Various iteration schemes are proposed by various authors to solve non-linear equations arising in the implementation of implicit Runge-Kutta methods. In this paper, a class of s-step non-linear scheme based on projection method is proposed to accelerate the convergence rate of those linear iteration schemes. In this scheme, sequence of numerical solutions is updated after each sub-step is completed. For 2-stage Gauss method, upper bound for the spectral radius of its iteration matrix was obtained in the left half complex plane. This result is extended to 3-stage and 4-stage Gauss methods by transforming the coefficient matrix and the iteration matrix to a block diagonal form. Finally, some numerical experiments are carried out to confirm the obtained theoretical results. Tue, 01 Jan 2019 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/3807 2019-01-01T00:00:00Z