Two-Stage Sand-Runge-Kutta Methods Powerful for Non-linear Equations with Multiple Solutions

A system of non-linear equations with multiple solutions is often appearing in field of science and technology. To solve efficiently such a system, the Runge-Kutta method is applied to an initial value problem of ordinary differential equations equivalent to its system. Then a class of iteration methods having third-order convergence for a simple solution is constructed. In addition, it is shown that there is a method possessing second-order convergence for a solution with any multiplicity in this class.