In this manuscript, an implicit cosimulation method is analyzed, where the solvers are coupled by algebraic constraint equations. We discuss cosimulation approaches on index-2 and on index-1 level and investigate constant, linear and quadratic approximation functions for the coupling variables. The key idea of the method presented here is to discretize the Lagrange multipliers between the macrotime points (extended multiplier approach) so that the coupling equations and their time derivatives can simultaneously be fulfilled at the macrotime points. Stability and convergence of the method are investigated in detail. Following the stability analysis for time integration schemes based on Dahlquist's test equation, an appropriate cosimulation test model is used to examine the numerical stability of the presented cosimulation method. Discretizing the cosimulation test model by means of a linear cosimulation approach yields a system of linear recurrence equations. The spectral radius of the recurrence equation system characterizes the numerical stability of the underlying cosimulation method. As for time integration methods, 2D stability plots are used to graphically illustrate the stability behavior of the coupling approach.