Abstract:
Various iteration schemes have been proposed to solve the non linear equations arising in the implementation of implicit Runge-Kutta
methods. In more general scheme, when applied to an s-stage Runge-Kutta
method, each step of the iteration requires s function evaluations and s sets
of linear equations to be solved. Convergence rates were obtained when
applied to the scalar differential equation 𝑥
′ = 𝑞𝑥. The convergence rate of
this scheme is further investigated by forcing the spectral radius 𝜌 𝑀 𝑧 of
the iteration matrix 𝑀 𝑧 to be zero at 𝑧 = 0, to be zero at 𝑧 = ∞ and to be
zero at 𝑧 = 0 and 𝑧 = ∞, where 𝑧 = ℎ𝑞 and ℎ is the fixed step-size. The
respective optimal parameters of the improved schemes are obtained for two
stage Gauss method. Numerical experiments are carried out to evaluate and
compare the efficiency of the new schemes and the original scheme.
