A two-stage difference scheme with complex coefficients for solving systems of the first order nonlinear ODE

A two-stage difference scheme with complex coefficients for solving systems of the first order nonlinear ordinary differential equations is presented. To derive the proposed scheme, a new approach is used. It is based on the scheme which differs from diagonal-implicit Runge–Kutta schemes. In a linear case the proposed scheme is equivalent to the Rosenbrock scheme with complex coefficients. The proposed scheme has the second order of truncation error, but considerably more accurate than the Rosenbrock scheme, provided the same time step is used. In case of equal computational requirements comparison with the Rosenbrock scheme shows that the new scheme generates more accurate solution for smooth components in stiff problems, utilizing a time mesh with two times steps. Numerical results are presented to illustrate accuracy of the proposed scheme.