The convergence of two Newton-like methods for solving block nonlinear equations and a class of r-point (r + 1)st-order A-stable one-block methods Original Research Article

In this paper, for a Newton-like method for solving block nonlinear equations arising in the numerical solution of stiff ODEs y′ = f(y), which involves a smaller quantity of computation, we prove that it is convergent and the convergence is independent of the stiffness of f(y), and give the error estimate. Furthermore, we present a modified Newton-like method involving an even smaller quantity of computation in certain cases, and prove that the modified method is convergent and the convergence is independent of the stiffness of f(y) for constant coefficient linear ODEs. Secondly, for any positive integer r, we discuss and construct a class of r-point(r + 1)st-order A-stable one-block methods suitable for the solution of stiff ODEs. Finally, we put forward an implementation strategy combining this one-block method and the r-point rth-order A-stable one-block method of Zhao Shuangsuo and Zhang Guofeng (1997). The numerical tests show that the strategy is efficient.