Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations

Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme

Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates

AbstractWe develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space

The time-domain numerical solution of Maxwell's electromagnetic equations, via the fourth order Runge-Kutta discontinuous Galerkin method.

This thesis presents a high-order numerical method for the Time-Domain solution of Maxwell's Electromagnetic equations in both one- and two-dimensional space. The thesis discuses the validity of high-order representation and improved boundary representation. The majority of the theory is concerned with the formulation of a high-order scheme which is capable of providing a numerical solution for specific two-dimensional scattering problems. Specifics of the theory involve the selection of a suitable numerical flux, the choice of appropriate boundary conditions, mapping between coordinate systems and basis functions. The effectiveness of the method is then demonstrated through a series of examples

The Discontinuous Galerkin Method for Maxwell\u27s Equations: Application to Bodies of Revolution and Kerr-Nonlinearities

Die unstetige Galerkinmethode (UGM) wird auf die rotationssymmetrischen und Kerr- Maxwell-Gleichungen angewandt. Essentiell ist hierbei der numerische Fluss. Für die rotationssymmetrischen Maxwell-Gleichungen wird ein exakter Fluss vorgestellt und unter Ausnutzung der Symmetrie der Aufwand reduziert. Für die Kerr-Maxwell-Gleichungen führt der exakte numerische Fluss auf eine ineffiziente UGM, weswegen approximative Flüsse miteinander verglichen werden. Wir erhalten optimale Konvergenz

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

International audienceA high-order leap-frog based non-dissipative discontinuous Galerkin time-domain 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 Nth-order leap-frog 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

An analysis of the Leap-Frog Discontinuous Galerkin method for Maxwell equations

In this paper, we explore the accuracy limits of a Finite-Element Time-Domain method applied to the Maxwell equations, based on a Discontinuous Galerkin scheme in space, and a Leap-Frog temporal integration. The dispersion and dissipation properties of the method are investigated, as well as the anisotropy of the errors. The results of this novel analysis are represented in a practical and comprehensible manner, useful for the application of the method, and for the understanding of the behavior of the errors in Discontinuous Gelerkin Time-Domain methods. A comparison with the Finite-Difference Time-Domain method, in terms of computational cost, is also includedThe work described in this paper and the research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013, under grant agreement no 205294 (HIRF SE project), and from the Spanish National Projects TEC2010-20841-C04-04, CSD2008-00068, and the Junta de Andalucia Project P09-TIC-