Preliminary Investigation of a Nonconforming Discontinuous Galerkin Method for Solving the Time-Domain Maxwell Equations (original) (raw)
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2007
This work is concerned with the design of a hp-like discontinuous Galerkin (DG) method for solving the 2D time-domain Maxwell's equations on non-conforming locally refined triangular meshes. The proposed DG method allows non-conforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme. It is an extension of the DG formulation recently studied in . After reviewing the stability properties of the DG method introduced in [18], we present a new implementation of this method which allows local h-refinement and p-enrichment and which is based on a numerical quadrature formula for the computation of the flux matrices associated to non-conforming interfaces. Numerical experiments are presented which both validate the theoretical results of and provide further insights regarding the practical performance of the resulting hp-like DG method, particulary when non-conforming locally refined meshes are employed.
Mathematics and Computers in Simulation, 2015
We present a high-order discontinuous Galerkin (DG) method for solving the time-dependent Maxwell equations on non-conforming hybrid meshes. The hybrid mesh combines unstructured tetrahedra for the discretization of irregularly shaped objects with a hexahedral mesh for the rest of the computational domain. The transition between tetrahedra and hexahedra is completely non-conform, that is, no pyramidal or prismatic elements are introduced to link these elements. Within each mesh element, the electromagnetic field components are approximated by a arbitrary order nodal polynomial and a centered approximation is used for the evaluation of numerical fluxes at inter-element boundaries. The time integration of the associated semi-discrete equations is achieved by a fourth-order leapfrog scheme. The method is described and discussed, including algorithm formulation, stability, and practical implementation issues such as the hybrid mesh generation and the computation of flux matrices with cubature rules. We illustrate the performance of the proposed method on several two-and three-dimensional examples involving comparisons with DG methods on single element-type meshes. The results show that the use of non-conforming hybrid meshes in DG methods allows for a notable reduction in computing time without sacrificing accuracy.
A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics
Journal of Computational and Applied Mathematics, 2010
In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell's equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leapfrog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method.
Numerical Mathematics: Theory, Methods and Applications, 2009
A high-order leapfrog based non-dissipative discontinuous Galerkin timedomain method for solving Maxwell's equations is introduced and analyzed. The proposed method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements, with a N th-order leapfrog time scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with high-order elements show the potential of the method.
2009
This work is concerned with the design of a hp-like discontinuous Galerkin (DG) method for solving the two-dimensional time-domain Maxwell equations on non-conforming locally refined triangular meshes. The proposed DG method allows non-conforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leapfrog time integration scheme. It is an extension of the DG formulation recently studied in [13]. Several numerical results are presented to illustrate the efficiency and the accuracy of the method, but also to discuss its limitations, through a set of 2D propagation problems in homogeneous and heterogeneous media.
Discontinuous Galerkin methods for Maxwell's equations in the time domain
Comptes Rendus Physique, 2006
In this article, we describe a new high-order Discontinuous Galerkin approach to Maxwell's equations in the time domain. This approach is based on hexahedral meshes and uses a mass-lumping technique. Thanks to the orthogonality of the basis functions and a judicious choice of the approximation spaces, it provides an efficient solver for these equations in terms of storage and CPU time. To cite this article: G.
Journal of Computational Physics, 2013
In this article, we present an Interior Penalty discontinuous Galerkin Time Domain (IPDGTD) method on non-conformal meshes. The motivation for a non-conformal IPDGTD comes from the fact there are applications with very complicated geometries (for example, IC packages) where a conformal mesh may be very difficult to obtain. Therefore, the ability to handle non-conformal meshes really comes in handy. In the proposed approach, we first decompose the computational domain into non-overlapping subdomains. Afterward, each sub-domain is meshed independently resulting in non-conformal domain interfaces, but simultaneously providing great flexibility in the meshing process. The non-conformal triangulations at sub-domain interfaces can be naturally supported within the IPDGTD framework. Moreover, a MPI parallelization together with a local time-stepping strategy is applied to significantly increase the efficiency of the method. Furthermore, a general balancing strategy is described. Through a practical example with multi-scale features, it is shown that the proposed balancing strategy leads to better use of the available computational resources and reduces substantially the total simulation time. Finally, numerical results are included to validate the accuracy and demonstrate the flexibilities of the proposed non-conformal IPDGTD.
Journal of Computational Physics, 2006
In this paper, we present a non-dissipative spatial high-order discontinuous Galerkin method to solve the Maxwell equations in the time domain. The non-intuitive choice of the space of approximation and the basis functions induce an important gain for mass, stiffness and jump matrices in terms of memory. This spatial approximation, combined with a leapfrog scheme in time, leads also to a fast explicit and accurate method. A study of the dispersive error is carried out and a stability condition for the proposed scheme is established. Some comparisons with other schemes are presented to validate the new scheme and to point out its advantages. Finally, in order to improve the efficiency of the method in terms of CPU time on general unstructured meshes, a strategy of local time-stepping is proposed.