Discontinuous Galerkin finite element methods for (non)conservative partial differential equations

Rhebergen, Sander

doi:10.3990/1.9789036529648

Cited by 4 publications

(8 citation statements)

References 71 publications

(185 reference statements)

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“…Additionally, the characteristics of A ijk at an interface affect the numerical flux (q k n j ) * at that interface. Derivation of an appropriate numerical flux which is stable and convergent can be done using techniques demonstrated in Hesthaven and Warburton [27] and Rhebergen [28].…”

Section: Non-conservation Form Fluxesmentioning

confidence: 99%

Physics-Based-Adaptive Plasma Model for High-Fidelity Numerical Simulations

Datta

Shumlak

2018

Front. Phys.

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Section: Non-conservation Form Fluxesmentioning

confidence: 99%

Physics-Based-Adaptive Plasma Model for High-Fidelity Numerical Simulations

Datta

Shumlak

2018

Front. Phys.

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“…

The discontinuous Galerkin finite element method (DGFEM) developed by Rhebergen et al [1] offers a robust method for solving systems of nonconservative hyperbolic partial differential equations but, as we show here, does not satisfactorily deal with topography in shallow water flows at lowest order (so-called DG0, or equivalently finite volume). In particular, numerical solutions of the space-DG0 discretised one-dimensional shallow water equations over varying topography are not truly 'well-balanced'.

…”

mentioning

confidence: 89%

“…We note also that, for DG1 and above, if b h is initially discontinuous across elements (cf. [1]) then it evolves to a nearby continuous solution, at which point numerical solutions for b and h remain steady. For completeness, a proof that the DG1 discretization satisfies all aspects of well-balanced flow, first published in RBV2008, is reproduced here using our notation in appendix A.…”

Section: Does Rest Flow Remain At Rest?mentioning

confidence: 99%

“…There exists a powerful class of numerical methods for solving hyperbolic problems (e.g., LeVeque [3]), often motivated by the need to capture shock formation which are a consequence of nonlinearities in the governing equations and manifest as discontinuities in the solutions. One such numerical scheme that can be applied to hyperbolic problems is the discontinuous Galerkin finite element method (DGFEM); the main aims of this work are (i) to highlight an unsatisfactory issue of the DGFEM scheme developed by Rhebergen et al [1] (hereon RBV2008), which concerns well-balancedness and arises when integrating the SWEs with varying topography at lowest order; and (ii) to give a comprehensive proof of the numerical artefact that causes it. Knowledge of this issue -overlooked in RBV2008 and hitherto unreported in detail -first arose in Kent et al [4], who commented on the problem but did not provide proof of the result.…”

Section: Introductionmentioning

confidence: 99%

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Ensuring 'well-balanced' shallow water flows via a discontinuous Galerkin finite element method: issues at lowest order

Kent¹,

Bokhove²

2020

Preprint

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The discontinuous Galerkin finite element method (DGFEM) developed by Rhebergen et al. [1] offers a robust method for solving systems of nonconservative hyperbolic partial differential equations but, as we show here, does not satisfactorily deal with topography in shallow water flows at lowest order (so-called DG0, or equivalently finite volume). In particular, numerical solutions of the space-DG0 discretised one-dimensional shallow water equations over varying topography are not truly 'well-balanced'. A numerical scheme is well-balanced if trivial steady states are satisfied in the numerical solution; in the case of the shallow water equations, initialised rest flow should remain at rest for all times. Whilst the free-surface height and momentum remain constant and zero, respectively, suggesting that the scheme is indeed well-balanced, the fluid depth and topography evolve in time. This is both undesirable and unphysical, leading to incorrect numerical solutions for the fluid depth, and is thus a concern from a predictive modelling perspective. We expose this unsatisfactory issue, both analytically and numerically, and indicate a solution that combines the DGFEM formulation for nonconservative products with a fast and stable well-balanced finite-volume method. This combined scheme bypasses the offending issue and successfully integrates nonconservative hyperbolic shallow water-type models with varying topography at lowest order. We briefly discuss implications for the definition of a well-balanced scheme, and highlight applications when higher-order schemes may not be desired, which give further value to our finding beyond its exposure alone.

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“…The last term on the left‐hand side of Eq. is the nonconservative product, where

G \in ℝ^{4 \times 2 \times 4}

is a third‐order tensor that cannot be expressed as the Jacobian of some matrix . Let G ijk be a component of

bold-italicG

.…”

Section: Problem Descriptionmentioning

confidence: 99%

An a priori error estimate for the local discontinuous Galerkin method applied to two‐dimensional shallow water and morphodynamic flow

Mirabito

Dawson

Aizinger

2014

Numerical Methods Partial

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The application of discontinuous Galerkin (DG) methods to the numerical solution of the two‐ and three‐dimensional shallow water equations has seen increased interest over the past decade. In this article, previous work by the second author and several collaborators on the application and analysis of the DG method is extended to coupled shallow water/bed morphology dynamics, also called morphodynamics. Morphodynamic boundary value problems are used to simultaneously model shallow water hydrodynamics and sediment transport processes in estuarine and coastal systems. The governing equations of interest arise when the two‐dimensional Saint‐Venant equations are tightly coupled to the corresponding Exner equation. The tight coupling of these two processes presents numerous modeling and analytical challenges. The resulting nonlinear system is incompletely parabolic, and contains a nonconservative product. In this work, some of these analytical challenges are tackled by applying a local discontinuous Galerkin method to the morphodynamic system; the quantities of interest and their gradients are solved separately. A general, semidiscrete finite element formulation is presented, and after justifying some mild assumptions, a new a priori error estimate for the system is derived. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 397–421, 2015

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Discontinuous Galerkin finite element methods for (non)conservative partial differential equations

Cited by 4 publications

References 71 publications

Physics-Based-Adaptive Plasma Model for High-Fidelity Numerical Simulations

Physics-Based-Adaptive Plasma Model for High-Fidelity Numerical Simulations

Ensuring 'well-balanced' shallow water flows via a discontinuous Galerkin finite element method: issues at lowest order

An a priori error estimate for the local discontinuous Galerkin method applied to two‐dimensional shallow water and morphodynamic flow

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