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Abstract A high-order accurate scheme for solving the time-domain Maxwell's equations with a generalized dispersive material model is described. The equations for the electric field are solved in second-order form, and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations (ADE). Numerical methods are developed for both second-order and fourth-order accuracy in space and time. The equations are discretized using finite-differences, and advanced in time with a single-stage, three-level, space–time scheme which remains stable up to the usual explicit CFL restriction, as proven using mode analysis. Because the equations are treated in their second-order form, there is no need for grid staggering, and instead a collocated grid is used. Composite overlapping grids are used to treat complex geometries with boundary-conforming grids, and a high-order upwind dissipation is added to ensure robust and stable approximations on overlapping grids. Numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes. Highlights • A novel scheme for dispersive Maxwell's equations in second-order form using auxiliary differential equations. • Arbitrary dispersive effects are treated through a recently developed generalized dispersive material (GDM) model. • A three time-level, fourth-order accurate and memory efficient scheme is derived that has a large CFL "one" time-step. • Composite overlapping grids are used for complex geometries with accurate treatment of curved boundaries. • The scheme is verified using newly developed exact solutions for the dispersive equations. [ABSTRACT FROM AUTHOR]

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw [1] how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy. [ABSTRACT FROM AUTHOR]

We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level modified equation approach for Maxwell's equations in second-order form for the electric field coupled to ODEs for the polarization vectors and population densities of the atomic levels. The resulting scheme has a large CFL-one time-step. Curved interfaces between different materials are accurately treated with curvilinear grids and compatibility conditions. A novel hierarchical modified equation approach leads to an explicit scheme that does not require any nonlinear iterations. The hierarchical approach at interfaces leads to local updates at the interface with no coupling in the tangential directions. Complex geometry is treated with overset grids. Numerical stability is maintained using high-order upwind dissipation designed for Maxwell's equations in second-order form. The scheme is carefully verified for a number of two and three-dimensional problems. The resulting numerical model with generalized dispersion and arbitrary nonlinear multi-level system can be used for many plasmonic applications such as for ab initio time domain modeling of nonlinear engineered materials for nanolasing applications, where nano-patterned plasmonic dispersive arrays are used to enhance otherwise weak nonlinearity in the active media. • A novel scheme for Maxwell's equations and nonlinear active media is developed. • The method is a single-step 4th-order accurate scheme with a large CFL-one time-step. • Active materials are treated with a general multi-level atomic system. • Overset grids used for complex geometries with boundary conforming grids. • Interface equations are solved efficiently with a novel hierarchical approach. [ABSTRACT FROM AUTHOR]

• A novel scheme for multi-material Maxwell's equations with general linear dispersive effects. • High-order accurate treatment of interfaces between dispersive materials using compatibility conditions. • A single stage, three level, fourth-order accurate scheme is developed with a large CFL-one time-step. • Composite overlapping grids are used for complex geometries with boundary conforming grids. • Interface equations are solved efficiently with a novel hierarchical approach. A high-order accurate scheme for solving the time-domain dispersive Maxwell's equations and material interfaces is described. Maxwell's equations are solved in second-order form for the electric field. A generalized dispersive material (GDM) model is used to represent a general class of linear dispersive materials and this model is implemented in the time-domain with the auxiliary differential equation (ADE) approach. The interior updates use our recently developed second-order and fourth-order accurate single-stage three-level space-time finite-difference schemes, and this paper extends these schemes to treat interfaces between different dispersive materials. Composite overlapping grids are used to treat complex geometry, with Cartesian grids generally covering most of the domain and local conforming grids representing curved boundaries and interfaces. Compatibility conditions derived from the interface jump conditions and governing equations are used to derive accurate numerical interface conditions that define values at ghost points. Although some compatibility conditions couple the equations for the ghost points in tangential directions due to mixed-derivatives, it is shown how to decouple the equations to avoid solving a larger system of equations for all ghost points on the interface. The stability of the interface approximations is studied with mode analysis and it is shown that the schemes retain close to a CFL-one time-step restriction. Numerical results are presented in two and three space dimensions to confirm the accuracy and stability of the schemes. The schemes are verified using exact solutions for a planar interface, a disk in two dimensions, and a solid sphere in three dimensions. [ABSTRACT FROM AUTHOR]