Transparent boundary conditions for a discontinuous Galerkin Trefftz method

Egger, Herbert; Kretzschmar, Fritz; Schnepp, Sascha; Tsukerman, Igor; Weiland, Thomas

doi:10.1016/j.amc.2015.06.026

Cited by 12 publications

(17 citation statements)

References 41 publications

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“…The spectral-like accuracy highlights the value of developing DG schemes for CED. However, previous generations of DG schemes for CED were not globally constraint-preserving (Hesthaven and Warburton [32], Cockburn, Li and Shu [28] Kretzschmar et al [34], Egger et al [30], Bokil et al [23]). This prevents a harmonious blending of the best attributes of DGTD schemes with the best attributes of FDTD schemes.…”

Section: I) Introductionmentioning

confidence: 99%

von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers

Balsara

Käppeli

2019

Journal of Computational Physics

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The time-dependent equations of computational electrodynamics (CED) are evolved consistent with the divergence constraints on the electric displacement and magnetic induction vector fields. Respecting these constraints has proved to be very useful in the classic finitedifference time-domain (FDTD) schemes. As a result, there has been a recent effort to design finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes that satisfy the same constraints and, nevertheless, draw on recent advances in higher order Godunov methods. This paper catalogues the first step in the design of globally constraint-preserving DGTD schemes. The algorithms presented here are based on a novel DG-like method that is applied to a Yee-type staggering of the electromagnetic field variables in the faces of the mesh. The other two novel building blocks of the method include constraint-preserving reconstruction of the electromagnetic fields and multidimensional Riemann solvers; both of which have been developed in recent years by the first author.The resulting DGTD scheme is linear, at least when limiters are not applied to the DG scheme. As a result, it is possible to carry out a von Neumann stability analysis of the entire suite of DGTD schemes for CED at orders of accuracy ranging from second to fourth. The analysis requires some simplifications in order to make it analytically tractable, however, it proves to be extremely instructive. A von Neumann stability analysis is a necessary precursor to the design of

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Section: I) Introductionmentioning

confidence: 99%

von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers

Balsara

Käppeli

2019

Journal of Computational Physics

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“…We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds.1 studied in [3]. Another Trefftz discontinuous Galerkin (DG) formulation for time-dependent electromagnetic problems formulated as first-order systems has been proposed in [25] and analysed in [24] in one space dimension; it has been extended to full three-dimensional Maxwell equations in [10,11,23].We mention here that, independently of the Trefftz approach, space-time finite elements for linear wave propagation problems, originally introduced in [21] (see also [15,22]), have been used in combination with DG formulations e.g. in [7,14,35] and, more recently, in [8,16,17,27].In this paper we extend the Trefftz-DG method of [24] to initial boundary value problems for the acoustic wave equations posed on Lipschitz polytopes in arbitrary dimensions.…”

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confidence: 99%

“…Remark 4.2 below. The Trefftz formulation for Maxwell's equations of [10,11,23,25] corresponds to the "unpenalised" version of that one proposed here (the numerical experiments in [24, §7.5] show that the numerical error depends very mildly on the penalisation parameters).We first describe the IBVP under consideration in §2, the assumptions on the mesh in §3 and the Trefftz-DG formulation in §4. Following the thread of [18,24], in §5.2 and §5.3 we prove that the scheme is well-posed, quasi-optimal, dissipative (quantifying dissipation using the jumps of the discrete solution), and derive error estimates for some traces of the solution on the mesh skeleton.…”

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confidence: 99%

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A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

Moiola

Perugia

2017

Numer. Math.

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We introduce a space-time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one dimensional scheme of Kretzschmar et al. (2016, IMA J. Numer. Anal., 36, 1599-1635. Test and trial discrete functions are space-time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeletonbased and mesh-independent norms. The space-time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on "tent-pitched" meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds.1 studied in [3]. Another Trefftz discontinuous Galerkin (DG) formulation for time-dependent electromagnetic problems formulated as first-order systems has been proposed in [25] and analysed in [24] in one space dimension; it has been extended to full three-dimensional Maxwell equations in [10,11,23].We mention here that, independently of the Trefftz approach, space-time finite elements for linear wave propagation problems, originally introduced in [21] (see also [15,22]), have been used in combination with DG formulations e.g. in [7,14,35] and, more recently, in [8,16,17,27].In this paper we extend the Trefftz-DG method of [24] to initial boundary value problems for the acoustic wave equations posed on Lipschitz polytopes in arbitrary dimensions. We write the acoustic wave problem as a first-order system, as it is originally derived from the linearised Euler equations, [6, p. 14]; we consider piecewise-constant wave speed, Dirichlet, Neumann and impedance boundary conditions. The main focus of this paper is on the a priori error analysis of the Trefftz-DG scheme.The DG formulation proposed can be understood as the translation to time-domain of the Trefftz-DG formulation for the Helmholtz equation of [18], which in turn is a generalisation of the Ultra Weak Variational Formulation (UWVF) of [5]. The DG numerical fluxes are upwind in time and centred with a special jump penalisation in space. Under a suitable choice of the numerical flux coefficients, combining the proposed formulation with standard discrete spaces and complementing it with suitable volume terms, one recovers the DG formulation of [35], cf. Remark 4.2 below. The Trefftz formulation for Maxwell's equations of [10,11,23,25] corresponds to the "unpenalised" version of that one proposed here (the numerical experiments in [24, §7.5] show that the numerical error depends very mildly on the penalisation parameters).We first describe the IBVP under consideration in §2, the assumptions on the mesh in §3 and the Trefftz-DG formulation in §4. Following the thread of [18,24], in §5.2 and §5.3 we prove that the scheme is well-posed, quasi-optimal, dissipative (quantifying dissipation using the jumps of the discrete solution), and derive error estimates for some traces of the solution on the mesh skeleton. ...

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“…First numerical experiments with Trefftz space-time dG methods were performed in [27]. Currently there is significant activity on the topic [12,20,13,24,21]. In particular in [12,20] a Trefftz space-time local dG method for the Maxwell equations, written as a first order system, resulting in a two-field formulation, has been analysed.…”

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confidence: 99%

A Trefftz Polynomial Space-Time Discontinuous Galerkin Method for the Second Order Wave Equation

Banjai¹,

Georgoulis²,

Lijoka³

2017

SIAM J. Numer. Anal.

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Abstract. A new space-time discontinuous Galerkin (dG) method 1 utilising special Trefftz polynomial basis functions is proposed and fully analysed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [17,18]. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behaviour of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions d = 1 and d = 2. Numerical experiments highlight the effectivness of the Trefftz method in problems with energy at high frequencies.1. Introduction. The use of non-polynomial basis functions in the context of finite element methods dates back at least to the early 1980s [3]. In recent years, there has been much interest in using non-polynomial basis functions to discretize wave propagation problems in the frequency domain [8,25]. A prominent example is the use of plane wave bases to solve the Helmholtz equation at high frequencies. The motivation behind this approach is to reduce the number of degrees of freedom per wavelength required to obtain accurate results. Thus obtained Trefftz methods have proved very successful in practice, hence it is a natural question to ask whether they can be extended and whether they can be equally successful in the time-domain.The most natural way of including space-time Trefftz basis functions is within the confines of a space-time discontinuous Galerkin (dG) method. In this work, we construct and analyse a new space-time interior penalty dG method for the second order wave equation, that can utilize Trefftz polynomials as local basis functions. The method discretizes the wave equation in primal form and is defined using space-time slabs to ensure solvability on each time-step, as well as to aid the presentation and the analysis. However, with minor modifications, completely unstructured space-time meshes are, in principle, possible in the proposed space-time dG framework. This results in a stable, dissipative scheme for general polynomial bases. For Trefftz basis, we prove quasi-optimality in the dG energy norm, which we show to be an upper bound for a standard space-time energy norm. Numerical results in the dG norm show that the theoretical estimates of the convergence order are optimal. Furthermore, the numerical results show that higher order schemes have excellent energy conservation properties and work well for systems with energy at high frequencies.Comparison with standard polynomial bases show that the same convergence order and approximation properties is obtained with considerably fewer degrees of freedom. Furthermore, the implementation is less expensive due to integration being restricted to the space-time skeleton.

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Transparent boundary conditions for a discontinuous Galerkin Trefftz method

Cited by 12 publications

References 41 publications

von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers

von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

A Trefftz Polynomial Space-Time Discontinuous Galerkin Method for the Second Order Wave Equation

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