Simulation of Dynamical Processes in Long Josephson Junctions: Computation of Current-Voltage Characteristics and Round Error Growth Estimation for a Second-Order Difference Scheme

Abstract

The fourth-order Runge–Kutta method is commonly used to compute the current-voltage characteristics of stacks of Josephson junctions. The calculations are performed for long time intervals, and the results are updated four times at each time step. To reduce the calculation time, this study suggests using a second-order explicit scheme instead of the Runge–Kutta method. Good results are obtained in particular calculations. For all $n$, estimates of $\left\| {{{G}^{n}}} \right\|$ ensuring the bounded growth of the round errors are proved, where $G$ is the layer-to-layer transition operator. A specific feature of the scheme under consideration is that its coefficients depend not only on the grid step size ratio $\gamma = \tau {\text{/}}h$ but also on $\tau$ ($\tau {\text{ and }}h$ are the grid step sizes in $t$ and $x$). It is proved that, for all $\gamma \leqslant 1$, the eigenvalues of the characteristic matrix are within the unit disc ($\left| {{{\lambda }_{j}}({{e}^{{i\phi }}})} \right| \leqslant 1$ for all $0 \leqslant \phi \leqslant 2\pi$) at a distance $O(\tau )$ from the unit circle. The estimation method developed in this study can be used in studying other numerical methods.