Explicit finite-difference methods for non-linear dynamic systems: Froudeʹs pendulum Original Research Article

Numerical schemes based on first-order methods and a second-order method are developed and analysed for solving the non-linear Froudeʹs Pendulum system. The second-order method is developed by taking a linear combination of three first-order methods. Though implicit in nature, with the resulting improvements in stability, the methods are applied explicitly. These methods are extended to a more general second-order ordinary differential equation. Stability of the numerical methods is approached from the point of view of dynamical systems. It is found that the methods are stable under the same conditions as those required by the linearized schemes in the neighbourhood of constant, stable, fixed points of the underlying initial-value problem. For the second-order method, this result is found to hold for any chosen value of the step-length. A major advantage of using the second-order method is the absence of solutions to the discrete finite-difference equation that do not correspond to the continuous dynamical system. This contrasts starkly with the Runge-Kutta methods which are found to produce solutions that converge to a false asymptote or wrong solutions in a deceptively smooth manner if the step-length is not rightly chosen. A selection of examples are carried out on the non-linear Froudeʹs pendulum system to illustrate the differing predictions obtained by the methods. Purchase $35.95