Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media
Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media
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DOI：
10.1016/j.jcp.2017.08.009
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发表时间：
2017-04
期刊：
影响因子：
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通讯作者：
V. Bokil;Yingda Cheng;Yan Jiang;Fengyan Li
中科院分区：
文献类型：
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作者：
V. Bokil;Yingda Cheng;Yan Jiang;Fengyan Li
电磁波在一般介质中的传播是由时间相关的麦克斯韦偏微分方程（PDE），再加上本构关系，描述了响应的媒体。在这项工作中，我们专注于非线性光学介质的响应是由一个一阶非线性常微分方程组（ODE），其中包括一个单一的共振线性洛伦兹色散，和非线性来自瞬时的电子克尔响应和残留的拉曼分子振动响应。为了设计高效、精确和稳定的计算方法，我们将高阶间断Galerkin空间离散化应用于混合偏微分方程-常微分方程麦克斯韦系统，并给出了几种数值通量的选择，所得到的半离散方法被证明是能量稳定的.在一定的非线性强度的限制下，也建立了误差估计。当我们转向全离散方法时，实现可证明稳定性的挑战在于非线性项的时间离散化。为了克服这一点，提出了新的策略来处理在我们的模型的框架内的二阶蛙跳和隐式梯形时间积分器的非线性。整体算法的性能通过数值模拟的扭结和反扭结波，和三次谐波产生的孤子传输。
The propagation of electromagnetic waves in general media is modeled by the time-dependent Maxwell's partial differential equations (PDEs), coupled with constitutive laws that describe the response of the media. In this work, we focus on nonlinear optical media whose response is modeled by a system of first order nonlinear ordinary differential equations (ODEs), which include a single resonance linear Lorentz dispersion, and the nonlinearity comes from the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. To design efficient, accurate, and stable computational methods, we apply high order discontinuous Galerkin discretizations in space to the hybrid PDE-ODE Maxwell system with several choices of numerical fluxes, and the resulting semi-discrete methods are shown to be energy stable. Under some restrictions on the strength of the nonlinearity, error estimates are also established. When we turn to fully discrete methods, the challenge to achieve provable stability lies in the temporal discretizations of the nonlinear terms. To overcome this, novel strategies are proposed to treat the nonlinearity in our model within the framework of the second-order leap-frog and implicit trapezoidal time integrators. The performance of the overall algorithms are demonstrated through numerical simulations of kink and antikink waves, and third-harmonic generation in soliton propagation.