Browsing by Author "Shanjeevan, T."
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Item Schemes with improving rate of convergence for three-stage Gauss method(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Shanjeevan, T.; Vigneswaran, R.The various iteration schemes have been proposed to solve the nonlinear equations arising in the implementation of s-stage implicit Runge-Kutta methods applied to solve a system of n ordinary differential equations with initial conditions () = () ; ≤ ≤ , ∶ [ , ] ⟶ ℝ, () = and ∶ ℝ ⟶ ℝ. A more general scheme, which was already proposed, is given by { ⊗ ( − ℎ ⊗ )} = ( ⊗ )() + ( ⊗ ), = + ( ⊗ ), = 1,2,3, …, where and are real non-singular parametric matrices, is a strictly lower triangular matrix, is the Jacobian evaluated at some recent point , ℎ is a fixed step size, and are identity matrices with order s and n respectively, ⊗ is the direct product of S with and is a real constant and () is the approximate diffect correction given by () = − + ℎ( ⊗ )() , where A is a coefficient matrix of the method, and () = () ⊕ () ⊕ () ⊕ … … ⊕ () is column vector. The rate of convergence of this scheme is examined when it is applied to the scalar differential equations = and the rate of convergence depends on the spectral radius [()] of the iteration matrix (), a function of = ℎ, where ℎ is a fixed step size. This scheme had already been investigated by assuming that () has only one non-zero eigen-value. In this problem, this scheme is further investigated by forcing [()] to be zero at = 0 and to be zero at = ∞ in addition to the constraint that () has only one non-zero eigenvalue. Results are obtained for three-stage Gauss method. A number of numerical experiments are carried out to confirm the results obtained for three stage Gauss method.