Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

Dedner, Andreas; Klöfkorn, Robert

doi:10.1007/s42967-021-00134-5

Cited by 3 publications

(9 citation statements)

References 49 publications

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“…where λ e is an estimate of the maximum wave speed on the interface e. Other options are implemented in Dune-Fem-DG (cf. [18,19]) as well.…”

Section: Dg-sem In Spacementioning

confidence: 94%

“…This superconvergence can be proven using the theory of dual consistent SBP methods [39]. Here it suffices to say that it is a consequence of the order of the quadrature and choosing the upwind numerical flux (19).…”

Section: Comparison Of Terminologymentioning

confidence: 99%

“…Internally, PDEs described in UFL are translated into C++ code just-in-time, to ensure that the resulting simulation code is performant. For a more detailed description we refer to [22,19] and the tutorial [20].…”

Section: Governing Equations and Simulation Softwarementioning

confidence: 99%

“…Note that B τ = e Nτ e ⊺ Nτ − e 1 e ⊺ 1 and (e ⊺ Nτ ⊗ I ξ )(u − u * ) = (u Nτ − u Nτ ) = 0. Using (19), the second term in (23) can therefore be expressed as…”

Section: Dg-sem and Lobatto Iiicmentioning

confidence: 99%

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Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

Versbach¹,

Linders²,

Klöfkorn³

et al. 2022

Preprint

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We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

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“…where λ e is an estimate of the maximum wave speed on the interface e. Other options are implemented in Dune-Fem-DG (cf. [18,19]) as well.…”

Section: Dg-sem In Spacementioning

confidence: 94%

Section: Comparison Of Terminologymentioning

confidence: 99%

Section: Governing Equations and Simulation Softwarementioning

confidence: 99%

“…Note that B τ = e Nτ e ⊺ Nτ − e 1 e ⊺ 1 and (e ⊺ Nτ ⊗ I ξ )(u − u * ) = (u Nτ − u Nτ ) = 0. Using (19), the second term in (23) can therefore be expressed as…”

Section: Dg-sem and Lobatto Iiicmentioning

confidence: 99%

See 2 more Smart Citations

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

Versbach¹,

Linders²,

Klöfkorn³

et al. 2022

Preprint

Self Cite

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“…On Ω 1 we discretize ( 27) in space using a 1st order finite volume discretization, implemented in DUNE [1,5]. The is grid shown in Figure 7.…”

Section: Gas Quenchingmentioning

confidence: 99%

Waveform Relaxation with asynchronous time-integration

Meisrimel¹,

Birken²

2021

Preprint

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We consider Waveform Relaxation (WR) methods for partitioned time-integration of surface-coupled multiphysics problems. WR allows independent time-discretizations on independent and adaptive time-grids, while maintaining high time-integration orders. Classical WR methods such as Jacobi or Gauss-Seidel WR are typically either parallel or converge quickly.We present a novel parallel WR method utilizing asynchronous communication techniques to get both properties. Classical WR methods exchange discrete functions after time-integration of a subproblem. We instead asynchronously exchange time-point solutions during time-integration and directly incorporate all new information in the interpolants. We show both continuous and time-discrete convergence in a framework that generalizes existing linear WR convergence theory. An algorithm for choosing optimal relaxation in our new WR method is presented.Convergence is demonstrated in two conjugate heat transfer examples. Our new method shows an improved performance over classical WR methods. In one example we show a partitioned coupling of the compressible Euler equations with a nonlinear heat equation, with subproblems implemented using the open source libraries DUNE and FEniCS.

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