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Application Of Operator Splitting To The Maxwell Equations Includinga Source Term

机译：算子分裂在包括源项在内的麦克斯韦方程组中的应用

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摘要

Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations w'(t) = Aw(t) + f(t), A ∈ R~(n×n) split into two subproblems w'_1(t) = A_1w_1(t) + f_1(t) and w'_2(t) = A_(2w2)(t) + f_2(t), A = A_1 + A_2, f = f_1 + f_2. First, expressions for the leading term of the local error are derived for the Strang-Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems w'_1 = Aw_1 and w'_2 = f (with the split-off source term f). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nedelec finite element discretizations, we test the Gautschi-Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error.
机译：基于时间相关的麦克斯韦方程组的数值解，我们考虑了微分方程w'（t）= Aw（t）+ f（t），A∈R〜（n×n）的线性系统的分裂方法两个子问题w'_1（t）= A_1w_1（t）+ f_1（t）和w'_2（t）= A_（2w2）（t）+ f_2（t），A = A_1 + A_2，f = f_1 + f_2 。首先，针对Strang-Marchuk和对称加权的顺序拆分方法，导出了局部误差的前导表达式。假设子问题得到精确解决，进行的分析确认了这两种方案的预期二阶全局精度。其次，对通过有限差分或有限元离散化的麦克斯韦方程组进行了几个相关的数值测试。一个有趣的情况是拆分为子问题w'_1 = Aw_1和w'_2 = f（带有分离源项f）。对于中心有限差分交错离散化，我们考虑了二阶分裂方案，并将它们与经典的Yee方案进行了比较，该方案具有损失项和源项。对于矢量Nedelec有限元离散化，我们测试了Gautschi-Krylov时间积分方案。与拆分源项结合使用，可生成每个拆分步骤均精确的拆分方案。因此，方案的时间积分误差仅由分割误差组成。

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