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

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.

Various iteration schemes are proposed by various authors to solve nonlinear 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.