The numerical solution of ordinary differential equations using implicit integration methods can be computationally demanding. Recently, efficient semi-implicit methods have gained attention, providing a good balance between precision and speed. They often possess one-dimensional implicitness which is usually resolved analytically; however, in cases when an analytical solution cannot be obtained, one can use simpler iterative algorithms than conventional Newton iterations. In this study, we investigate the possibility of using fixed point iterations and evaluate their convergence to an analytical solution compared with classical Newton’s method. The obtained results clarify the properties of semi-implicit integrators and can improve their application to the design of simulation software.