Accelerating convergence rate of linear iteration schemes based on projection method for three-stage gauss method

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.