Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions

Fisher, Travis C.; Carpenter, Mark H.; Nordström, Jan; Yamaleev, Nail K.; Swanson, C. A.

doi:10.1016/j.jcp.2012.09.026

Cited by 151 publications

(181 citation statements)

References 32 publications

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“…With notation (3.13), the nodal values of the derivative of a nodal approximation w are given by 14) or in index notation…”

Section: Polynomial Approximation On the Reference Elementmentioning

confidence: 99%

“…The new flux functions, denoted with an overbar, can be viewed as subcell fluxes on a complementary staggered subcell grid [14,19]. The contravariant flux functions on the complementary grid remain consistent and high-order when they are computed according to [18, Section 4.5 and Appendix A.3]…”

Section: Furthermore the Sbp-property Can Be Used To Obtain Alternatmentioning

confidence: 99%

“…This is problematic as it is not obvious that discretisations of the split form remain conservative, yet conservation is desired for the numerical solution to exhibit correct shock speeds. Recent success has been had using diagonal norm summation-by-parts (SBP) finite difference operators to discretise the spatial derivatives in the split formulation of the equations [8,14,15,16]. Fisher et al [14] show that split form operators derived from SBP derivative matrices are consistent and conservative in the Lax-Wendroff sense.…”

Section: Introductionmentioning

confidence: 99%

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An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Wintermeyer

Winters

Gassner

et al. 2017

Journal of Computational Physics

102

114

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We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

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“…With notation (3.13), the nodal values of the derivative of a nodal approximation w are given by 14) or in index notation…”

Section: Polynomial Approximation On the Reference Elementmentioning

confidence: 99%

Section: Furthermore the Sbp-property Can Be Used To Obtain Alternatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

Wintermeyer

Winters

Gassner

et al. 2017

Journal of Computational Physics

102

114

View full text Add to dashboard Cite

show abstract

“…These grids allow the finite difference operations to be written as simple flux differences, analogous to the approach of the finite volume method. In a previous paper [15], we showed that this telescopic flux difference form yields a generalized summation-by-parts (SBP) property that is used to show that the weak solution to 2.1 is recovered when the solution converges.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Following Hauke et al [14] this is shown by casting the viscous flux in a quasi-linear form, 15) where c(q, x) may be a scalar constant, a variable matrix that depends on spatial location, or nonlinear in the conservation variable. An entropy function can be chosen such that the viscous coefficients are symmetric and positive semi-definite,…”

Section: Entropy Analysismentioning

confidence: 99%

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

Fisher

Carpenter

2013

Journal of Computational Physics

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289

265

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A kinetic energy preserving nodal discontinuous Galerkin spectral element method

Gassner

2014

Numerical Methods in Fluids

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SUMMARYIn this work, we discuss the construction of a skew‐symmetric discontinuous Galerkin (DG) collocation spectral element approximation for the compressible Euler equations. Starting from the skew‐symmetric formulation of Morinishi, we mimic the continuous derivations on a discrete level to find a formulation for the conserved variables. In contrast to finite difference methods, DG formulations naturally have inter‐domain surface flux contributions due to the discontinuous nature of the approximation space. Thus, throughout the derivations we accurately track the influence of the surface fluxes to arrive at a consistent formulation also for the surface terms. The resulting novel skew‐symmetric method differs from the standard DG scheme by additional volume terms. Those volume terms have a special structure and basically represent the discretization error of the different product rules. We use the summation‐by‐parts (SBP) property of the Gauss–Lobatto‐based DG operator and show that the novel formulation is exactly conservative for the mass, momentum, and energy. Finally, an analysis of the kinetic energy balance of the standard DG discretization shows that because of aliasing errors, a nonzero transport source term in the evolution of the discrete kinetic energy mean value may lead to an inconsistent increase or decrease in contrast to the skew‐symmetric formulation. Furthermore, we derive a suitable interface flux that guarantees kinetic energy preservation in combination with the skew‐symmetric DG formulation. As all derivations require only the SBP property of the Gauss–Lobatto‐based DG collocation spectral element method operator and that the mass matrix is diagonal, all results for the surface terms can be directly applied in the context of multi‐domain diagonal norm SBP finite difference methods. Numerical experiments are conducted to demonstrate the theoretical findings. Copyright © 2014 John Wiley & Sons, Ltd.

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Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions

Cited by 151 publications

References 32 publications

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains

A kinetic energy preserving nodal discontinuous Galerkin spectral element method

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