首页|Energy Stable Nodal DG Methods for Maxwell's Equations of Mixed-Order Form in Nonlinear Optical Media

Energy Stable Nodal DG Methods for Maxwell's Equations of Mixed-Order Form in Nonlinear Optical Media

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In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equa-tions considered previously in Bokil et al.(J Comput Phys 350:420-452,2017)and Lyu et al.(J Sci Comput 89:1-42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell's equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlin-ear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is fur-ther applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with dE as the number of the components of the electric field,only a dE × dE nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small non-linear systems are completely decoupled over one time step,rendering very high paral-lel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accu-racy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

Maxwell's equationsKerr and RamanDiscontinuous Galerkin methodEnergy stability

Maohui Lyu、Vrushali A.Bokil、Yingda Cheng、Fengyan Li

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State Key Laboratory of Scientific and Engineering Computing(LSEC),Academy of Mathematics and System Sciences,Chinese Academy of Sciences,Beijing 100190,China

Department of Mathematics,College of Science,Oregon State University,Corvallis,OR 97331,USA

Department of Mathematics,Michigan State University,East Lansing,MI 48824,USA

Department of Computational Mathematics,Science and Engineering,Michigan State University,East Lansing,MI 48824,USA

Department of Mathematical Sciences,Rensselaer Polytechnic Institute,Troy,NY 12180,USA

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中国博士后科学基金国家自然科学基金国家自然科学基金国家自然科学基金国家自然科学基金国家自然科学基金国家自然科学基金国家自然科学基金

2020TQ03441187113912101597DMS-1720116DMS-2012882DMS-2011838DMS-1719942DMS-1913072

2024

10.1007/s42967-022-00212-2

应用数学与计算数学学报
上海大学

应用数学与计算数学学报

影响因子：0.165
ISSN：1006-6330
年,卷(期)：2024.6(1)
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