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Register a new userDispersion Analysis of Finite Difference and Discontinuous Galerkin Schemes for Maxwells Equations in Linear Lorentz Media
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In this paper, we consider Maxwells 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 studied in our previous research [5,6]. 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. The results for the numerical dispersion analysis 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 Maxwells 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.
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In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.
We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwells equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level modified equation approach for Maxwells 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 Maxwells 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.
Motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction, we introduce locally bounded transport schemes for (partially-)continuous finite element spaces. The underlying high-order transport scheme is constructed by injecting the partially-continuous field into an embedding discontinuous finite element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and projecting back into the partially-continuous space; we call this an embedded DG scheme. We prove that this scheme is stable in L2 provided that the underlying upwind DG scheme is. We then provide a framework for applying limiters for embedded DG transport schemes. Standard DG limiters are applied during the underlying DG scheme. We introduce a new localised form of element-based flux-correction which we apply to limiting the projection back into the partially-continuous space, so that the whole transport scheme is bounded. We provide details in the specific case of tensor-product finite element spaces on wedge elements that are discontinuous P1/Q1 in the horizontal and continuous P2 in the vertical. The framework is illustrated with numerical tests.
In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order aposteriori sub-cell ADER-WENO finite volume emph{limiter}. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in cite{Dumbser2014}, the discrete solution within the troubled cells is textit{recomputed} by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.
This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the $(2k+1)$th order superconvergence for the cell averages, and the numerical traces in the discrete $L^2$ norm. In addition, superconvergence of order $k+2$ and $k+1$ are obtained for the error and its derivative at generalized Radau points. All theoretical findings are confirmed by numerical experiments.
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