Numerical instability in the breathing-mode dynamics of elastic rings Original Research Article

The method of lines is used to compute the breathing-mode dynamics of two-dimensional elastic rings. An explicit Runge-Kutta package is used as the time integrator with the spatial derivatives approximated by four different schemes, one spectral and three finite difference. For each spatial scheme, the performance of the Runge-Kutta package is evaluated for third, fifth and eighth order accurate Runge-Kutta methods, based on (2, 3), (4, 5) and (7, 8) Runge-Kutta pairs, respectively. It is shown that the inclusion of shear deformation ultimately leads to loss of all accuracy. With shear deformation neglected, blow up does not occur and the accuracy obtained is that of the underlying spatial differencing scheme.