Simulation of dynamic processes in long Josephson junctions. The problem on calculating the current–voltage characteristics. Numerical method for estimating the round-off error growth rate

As a rule, current–voltage characteristics are numerically calculated using the fourth-order Runge–Kutta scheme. The calculations are carried out over large time intervals and, at each time step, the right-hand sides of the equations are recalculated four times. To reduce the computation time, it is proposed to replace the Runge–Kutta scheme by an “explicit” second-order scheme. For $\tau=h$ and all $n$, estimations of $\|G^n\|$ guaranteeing the boundedness of the round-off error growth rate were proved, where $G$ is the operator of the transition from layer to layer and $\tau$, $h$ are the grid steps in $t$, $x$, respectively. In this work a numerico-analytical algorithm for estimating the round-off errors is developed for all $\tau\le h$. Their boundedness over the entire interval of calculating the current–voltage characteristics of long Josephson junctions when using the proposed scheme is proved. The calculations were carried out on a “Govorun” supercomputer with the REDUCE system.