Study of a Lyapunov-based discretization method using the midpoint approach

We propose a modification of an existing discretization technique. The technique we atempt to improve is a discretization method designed for homogeneous systems, characterized by first-order consistency, numerical convergence, and the noteworthy feature of preserving both the stability of the system’s equilibrium point and the type of convergence of the solutions toward the origin. Its design is based on the projection of the dynamics of the system over a level surface of the Lyapunov function, and the complementary use of the explicit Euler method. In this work the Euler’s explicit method is replaced by the midpoint method, and it is proved that, with such a change, the discretization methodology continues to preserve both the Lyapunov function and the type of convergence exhibited by the solutions of the original system. Furthermore, it is proven that this new approach improves the accuracy of the method since it can achieve second-order convergence.