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

Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisation of the Maxwell eqations

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 to fixed-order schemes. This comes without a significant increase in the computation 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 complete scheme. The analysis also provides practical information on the convergence of the dissipation and dispersion error, which is important when studying wave propagation phenomena

Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions

Time-integration methods for finite element discretisations of the second-order Maxwell equation

This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps

Space–time residual distribution on moving meshes

This article investigates the potential for an r-adaptation algorithm to improve the efficiency of space–time residual distribution schemes in the approximation of time-dependent hyperbolic conservation laws, e.g. scalar advection, shallow water flows, on unstructured, triangular meshes. In this adaptive framework the connectivity of the mesh, and hence the number of degrees of freedom, remain fixed, but the mesh nodes are continually “relocated” as the flow evolves so that features of interest remain resolved as they move within the domain. Adaptive strategies of this type are well suited to the space–time residual distribution framework because, when the discrete representation is allowed to be discontinuous in time, these algorithms can be designed to be positive (and hence stable) for any choice of time-step, even on the distorted space–time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the time-step, which can instead be chosen according to accuracy requirements. The order of accuracy of the fixed-mesh scheme is retained on the moving mesh in the majority of applications tested. Space–time schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law, so moving the mesh does not artificially change the flow volume for pure conservation laws. For shallow water flows over variable bed topography, the so-called C-property (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved. The r-adaptation is applied within each time-step by interleaving the iterations of the nonlinear solver with updates to mesh node positions. The node movement is driven by a monitor function based on weighted approximations of the scaled gradient and Laplacian of the local solution and regularised by a smoothing iteration. Numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed which show that, even for this simple implementation of the mesh movement, reductions in cpu times of up to 60% can be attained without increasing the error

High-order accurate discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations

We introduce a high-order accurate discontinuous Galerkin (DG) method for the indefinite frequency-domain Maxwell equations in three spatial dimensions. The novelty of the method lies in the way the numerical flux is computed. Instead of using the more popular local discontinuous Galerkin (LDG) or interior-penalty discontinuous Galerkin (IP-DG) numerical fluxes, we opt for a formulation which makes use of the local lifting operator. This allows us to choose a penalty parameter that is independent of the mesh size and the polynomial order. Moreover, we use a hierarchic construction of $H$(curl)-conforming basis functions, the first-order version of which correspond to the second family of Nédélec elements. We also provide a priori error bounds for our formulation, and carry out three-dimensional numerical experiments to validate the theoretical results

Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation

This article compares the discontinuous Galerkin finite element method (DG-FEM) with the $H(\mathrm{curl})$-conforming FEM in the discretisation of the second-order time-domain Maxwell equations with possibly nonzero conductivity term. While DG-FEM suffers from an increased number of degrees of freedom compared with $H(\mathrm{curl})$-conforming FEM, it has the advantage of a purely block-diagonal mass matrix. This means that, as long as an explicit time-integration scheme is used, it is no longer necessary to solve a linear system at each time step -- a clear advantage over $H(\mathrm{curl})$-conforming FEM. It is known that DG-FEM generally favours high-order methods whereas $H(\mathrm{curl})$-conforming FEM is more suitable for low-order ones. The novelty we provide in this work is a direct comparison of the performance of the two methods when hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$. The motivation behind this choice of basis functions is its growing importance in the development of $p$- and $hp$-adaptive FEMs.\ud \ud The fact that we allow for nonzero conductivity requires special attention with regards to the time-integration methods applied to the semi-discrete systems. High-order polynomial basis warrants the use of high-order time-integration schemes, but existing high-order schemes may suffer from a too severe time-step stability restriction as result of the conductivity term. We investigate several alternatives from the point of view of accuracy, stability and computational work. Finally, we carry out a numerical Fourier analysis to study the dispersion and issipation properties of the semi-discrete DG-FEM scheme and several of the time-integration methods. It is instructive in our approach that the dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps