Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media
Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media
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DOI：
10.1016/j.jcp.2019.05.022
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发表时间：
2018-10
期刊：
影响因子：
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通讯作者：
Yan Jiang;Puttha Sakkaplangkul;V. Bokil;Yingda Cheng;Fengyan Li
中科院分区：
文献类型：
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作者：
Yan Jiang;Puttha Sakkaplangkul;V. Bokil;Yingda Cheng;Fengyan Li
本文考虑了线性色散介质中的麦克斯韦方程组，该方程组由单极洛伦兹模型描述。本文研究了两类常用的空间离散方法：空间任意偶数阶精度的有限差分方法（FD）和高空间阶间断Galerkin有限元方法（DG）。这两种类型的空间离散耦合二阶半隐式蛙跳和隐式梯形时间格式。通过对半离散和全离散格式进行详细的色散分析，我们得到了洛伦兹色散补偿的色散误差的严格量化。特别地，可以考虑到模型参数和两种类型的方案的设计中的网格尺寸来进行色散误差的比较。本文的工作是我们以前对非线性色散光学介质能量稳定数值格式研究的继续[6]，[7]。本文所考虑的简化线性模型的数值频散分析结果，可以指导我们在更复杂和非线性模型的离散化参数的最佳选择。对色散麦克斯韦模型的全离散FD和DG格式的数值色散分析表明，数值色散误差与空间和时间离散方式、精度阶数、网格离散参数和模型参数有关。这里得到的结果不能通过考虑自由空间中的麦克斯韦方程的离散化而得到。特别是，我们的研究结果对比的优点和缺点，使用高阶FD或DG计划和蛙跳或梯形时间积分器在不同的频率范围内使用各种措施的数值色散误差。最后，我们强调了二阶精确时间离散化的局限性。
In this paper, we consider Maxwell's equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6], [7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell's equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors. Finally, we highlight the limitations of the second order accurate temporal discretizations considered.