Schemes with improving rate of convergence for three-stage Gauss method

Abstract

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.