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Fully Discrete Energy Stable Methods for Maxwell's Equations in Nonlinear Media

Banff International Research Station for Mathematical Innovation and Discovery

The propagation of electromagnetic waves is modeled by time-dependent Maxwell's equations coupled with constitutive laws that describe the response of the media. In this work, we examine a nonlinear optical model that describes electromagnetic waves in linear Lorentz and nonlinear Kerr and Raman media. To design efficient, accurate, and stable computational methods, we apply high order discontinuous Galerkin discretizations and finite difference schemes in space. The challenge to achieve provable stability for fully-discrete methods lies in the temporal discretizations of the nonlinear terms. To overcome this, novel modification is proposed for the second-order leap-frog and implicit trapezoidal time integrators. The performance of the method is demonstrated via numerical examples.

Mathematics; Numerical analysis; Partial differential equations; Women and early careers in math

video/mp4

Author affiliation: Rensselaer Polytechnic Institute

2019-11-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72253

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/