ABSTRACT

This paper deals with a difference scheme of second-order approximation using Laquerre transform for the one-dimensional Maxwell equations. Supplementary parameters are introduced into this difference scheme. These parameters are obtained by minimizing the error of a difference approximation for a Helmholtz equation. The optimal parameters do not depend on the step size and the number of nodes in the difference scheme. It is shown that the use of the Laguerre decomposition method allows obtaining higher accuracy of approximation of the equations in comparison with similar difference schemes when using the Fourier decomposition method. The second-order finite difference scheme with the parameters is compared to a fourth-order difference scheme in two cases: The use of the optimal difference scheme when solving a problem of electromagnetic pulse propagation in an inhomogeneous medium yields a solution accuracy comparable to that obtained with the fourth-order difference scheme. When solving an inverse problem the second-order difference scheme allows obtaining higher solution accuracy as compared to the fourth-order difference scheme. In these problems the second-order difference scheme with the supplementary parameters has decreased the calculation time by 20–25% as compared to the fourth-order difference scheme.

中文翻译：

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一维麦克斯韦方程组的有限差分格式

摘要

本文针对一维麦克斯韦方程组，采用拉奎尔（Laquerre）变换来处理二阶近似的差分格式。补充参数被引入该差异方案中。通过最小化亥姆霍兹方程的差分近似误差来获得这些参数。最佳参数不取决于步长和差分方案中的节点数。结果表明，与使用傅里叶分解方法的相似差分方案相比，使用拉盖尔分解方法可以获得更高的方程近似精度。在两种情况下，将具有参数的二阶有限差分方案与四阶差分方案进行比较：当解决电磁脉冲在不均匀介质中传播的问题时，使用最佳差分方案可产生与四阶差分方案相当的求解精度。当解决反问题时，与四阶差分方案相比，二阶差分方案允许获得更高的求解精度。在这些问题中，与四阶差分方案相比，带有补充参数的二阶差分方案使计算时间减少了20–25％。