Browsing by Author "Vigneswaran, R."
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Item Accelerating the rate of convergence of some efficient schemes for two-stage Gauss method(Faculty of Science, University of Kelaniya, Sri Lanka, 2016) Chamaleen, D.B.D.; Vigneswaran, R.The non-linear equations obtaining from the implicit s – stage Runge-Kutta methods have been solved by various iteration schemes. A scheme has been developed, which is computationally more efficient and avoids expensive vector transformations. The rate of convergence of this scheme is examined when it is applied to the scalar test differential equation = and the convergence rate depends on the spectral radius [()] of the iteration matrix (), where = ℎ and ℎ is the step-size. In this scheme, supremum of a lower bound for [()] is minimized over the left half - plane with the constraints requiring super-linear convergence at = 0 and → ∞ .Two new schemes with parameters are obtained for the two-stage Gauss-method. Numerical experiments are carried out in order to evaluate and compare the efficiency of the new schemes and the original scheme. Consider an initial value problem for stiff system of ordinary differential equations = (), () = , : ℝ → ℝ. An s-stage implicit Runge-Kutta method computes an approximation to the solution x () at discrete point = + ℎ by = + ℎ Σ ( ), where , ,…,, satisfy sn equations + ℎ , ), = 1,2, . . . , . = is the real coefficient matrix and = [ , ,…,] is the column vector of the Runge-Kutta method. Let = ⊕ ⊕ … ⊕ ∈ ℝand () = () ⊕ () ⊕ … ⊕ () ∈ ℝ. Then the above equation in , ,…, may be written by = ⊗ + ℎ( ⊗ )(), where = (1,1, … ,1) and ( ⊗ ) is the tensor product of the matrix with × identity matrix . The efficient scheme, which has been already proposed, is given by [ ⊗ ( − ℎ)] = ( ⊗ )( ⊗ – ) + ( ⊗ )( ⊗ – ) + ℎ( ⊗ )() + ℎ( ⊗ )(), = 1,2, …, In this scheme, supremum of a lower bound for [()] is minimized over ℂ, where ℂ = { ∈ / () ≤ 0 } with the constraints [()] = 0 at = 0 and [()] = 0 at → ∞. The parameters for the two-stage Gauss method are obtained and Numerical experiments are carried out.Item A class of s-step non-linear iteration scheme based on projection method for s-stage Runge-Kutta method.(International Research Symposium on Pure and Applied Sciences, 2017 Faculty of Science, University of Kelaniya, Sri Lanka., 2017) Kajanthan, S.; Vigneswaran, R.A variety of linear iteration schemes with reduced linear algebra costs have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods as an alternative to the modified Newton iteration scheme. 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. The s-step scheme is given. where is a scalar, O the zero vector. 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 this scheme. The non-singular matrix Q should be chosen to minimize the maximum of the spectral radius of the iteration matrix over the left half complex plane. For 2-stage Gauss method, upper bound for the spectral radius of the iteration matrix was obtained in the left half complex plane. In this approach, it is difficult to handle the 3-stage Gauss method and 4-stage Gauss methods. We transform the coefficient matrix and the iteration matrix to a block diagonal matrix. The result for s=2 is applied to other methods when s>2. Finally, some numerical experiments are carried out to confirm the obtained theoretical results. 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.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.