Efficient Lie derivative algorithm for two special nonlinear equations

This paper explores the effectiveness of the Lie derivative discretisation scheme applied to two particular types of nonlinear dynamical equations, both of which have the characteristic of time variables in the denominator position. The discrete structure of non-autonomous systems is established. In particular, we exclude time variables as state variables to prevent non-autonomous systems from becoming autonomous systems. Using this method, we compute the numerical solution of the system above and compare it with the precise solution and the numerical findings of Runge–Kutta, demonstrating the broad applicability of the Lie derivative numerical algorithm. Finally, we determine the CPU consumption time of two numerical algorithms, thus providing evidence of the high efficiency of the Lie derivative numerical algorithm.