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.

中文翻译：

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长约瑟夫森结的动力学过程仿真：电流电压特性的计算和二阶差分方案的舍入误差增长估计

摘要

四阶Runge–Kutta方法通常用于计算约瑟夫森结的堆栈的电流-电压特性。计算是在较长的时间间隔内进行的，并且每个时间步长将结果更新四次。为了减少计算时间，本研究建议使用二阶显式方案代替Runge-Kutta方法。在特定的计算中获得了良好的结果。对于所有\（n \），\（\ left \ | {{{G} ^ {n}}} \ right \ | \）的估计确保了舍入误差的有界增长，其中\（G \）是层到层过渡运算符。所考虑方案的一个特定特征是其系数不仅取决于网格步长比\（\ gamma = \ tau {\ text {/}} h \）而且在\（\ tau \）（\（\ tau {\ text {和}} h \）上也是\（t \）和\（x \）的网格步长。证明了，对于所有\（\ gamma \ leqslant 1 \），特征矩阵的特征值均在单位圆盘内（\（\ left | {{{\ lambda __ {j}}（{{e} ^ {{I \披}}}）} \右| \ leqslant 1 \）对所有\（0 \ leqslant \披\ leqslant 2 \ PI \） ）以一定距离\（O（\ TAU）\）从单位圆。本研究开发的估计方法可用于研究其他数值方法。