| 1. |
- Gassner, Gregor J, et al.
(författare)
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A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations
- 2016
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 272:2, s. 291-308
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we design an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations. The novel method uses a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, we show that combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. Finally, we prove that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced. Numerical tests are performed to demonstrate the theoretical findings.
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| 2. |
- Gassner, Gregor J, et al.
(författare)
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Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations
- 2016
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 327, s. 39-66
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Tidskriftsartikel (refereegranskat)abstract
- Fisher and Carpenter (High-order entropy stable finite difference schemes for non-linear conservation laws: Finite domains, Journal of Computational Physics, 252:518–557, 2013) found a remarkable equivalence of general diagonal norm high-order summation-by- parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the sub-cell finite volume flux. We show that besides the construction of entropy stable high order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we show we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor-Green vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.
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| 3. |
- Gassner, Gregor J, et al.
(författare)
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The BR1 scheme is stable for the compressible Navier–Stokes equations
- 2018
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 77:1, s. 154-200
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Tidskriftsartikel (refereegranskat)abstract
- We show how to modify the original Bassi and Rebay scheme (BR1) [F. Bassi and S. Rebay, A High Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, Journal of Computational Physics, 131:267–279, 1997] to get a provably stable discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss-Lobatto (GL) nodes for the compressible Navier-Stokes equations (NSE) on three dimensional curvilinear meshes.Specifically, we show that the BR1 scheme can be provably stable if the metric identities are discretely satisfied, a two-point average for the metric terms is used for the contravariant fluxes in the volume, an entropy conserving split form is used for the advective volume integrals, the auxiliary gradients for the viscous terms are computed from gradients of entropy variables, and the BR1 scheme is used for the interface fluxes.Our analysis shows that even with three dimensional curvilinear grids, the BR1 fluxes do not add artificial dissipation at the interior element faces. Thus, the BR1 interface fluxes preserve the stability of the discretization of the advection terms and we get either energy stability or entropy-stability for the linear or nonlinear compressible NSE, respectively.
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| 4. |
- Kopriva, David A, et al.
(författare)
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A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes
- 2016
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Ingår i: Computers & Fluids. - : Elsevier. - 0045-7930 .- 1879-0747. ; 139, s. 148-160
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Tidskriftsartikel (refereegranskat)abstract
- We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free- stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results.
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| 5. |
- Kopriva, David A., et al.
(författare)
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Error boundedness of discontinuous Galerkin spectral element approximations of hyperbolic problems
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We examine the long time error behavior of discontinuous Galerkin spectral element approximations to hyperbolic equations. We show that the choice of numerical flux at interior element boundaries affects the growth rate and asymptotic value of the error. Using the upwind flux, the error reaches the asymptotic value faster, and to a lower value than a central flux gives, especially for low resolution computations. The differences in the error caused by the numerical flux choice decrease as the solution becomes better resolved.
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| 6. |
- Kopriva, David A., et al.
(författare)
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Error Boundedness of Discontinuous Galerkin Spectral Element Approximations of Hyperbolic Problems
- 2017
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 72:1, s. 314-330
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Tidskriftsartikel (refereegranskat)abstract
- We examine the long time error behavior of discontinuous Galerkin spectral element approximations to hyperbolic equations. We show that the choice of numerical flux at interior element boundaries affects the growth rate and asymptotic value of the error. Using the upwind flux, the error reaches the asymptotic value faster, and to a lower value than a central flux gives, especially for low resolution computations. The differences in the error caused by the numerical flux choice decrease as the solution becomes better resolved.
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| 7. |
- Kopriva, David A., et al.
(författare)
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On the theoretical foundation of overset grid methods for hyperbolic problems : Well-posedness and conservation
- 2022
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Ingår i: Journal of Computational Physics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9991 .- 1090-2716. ; 448
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Tidskriftsartikel (refereegranskat)abstract
- We use the energy method to study the well-posedness of initial-boundary value problems approximated by overset mesh methods in one and two space dimensions for linear constant-coefficient hyperbolic systems. We show that in one space dimension, for both scalar equations and systems of equations, the problem where one domain partially oversets another is well-posed when characteristic coupling conditions are used. If a system cannot be diagonalized, as is usually the case in multiple space dimensions, then the energy method does not give proper bounds in terms of initial and boundary data. For those problems, we propose a novel penalty approach. We show, by using a global energy that accounts for the energy in the overlap region of the domains, that under well-defined conditions on the coupling matrices the penalized overset domain problems are energy bounded, conservative, well-posed and have solutions equivalent to the original single domain problem.
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| 8. |
- Kopriva, David A., et al.
(författare)
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On the theoretical foundation of overset grid methods for hyperbolic problems II : Entropy bounded formulations for nonlinear conservation laws
- 2022
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 471
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Tidskriftsartikel (refereegranskat)abstract
- We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes a two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. In addition to the interface coupling, which is required, entropy dissipation and coupling can optionally be added through the use of linear penalties within the overlap region.
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| 9. |
- Kopriva, David A., et al.
(författare)
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Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps
- 2021
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 88:1
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Tidskriftsartikel (refereegranskat)abstract
- We use the behavior of the L2 norm of the solutions of linear hyperbolic equations withdiscontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkinspectral element methods (DGSEM). Although the L2 norm is not bounded in terms of theinitial data for homogeneous and dissipative boundary conditions for such systems, the L2norm is easier to work with than a norm that discounts growth due to the discontinuities. Weshow that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot(or conservation) condition has the same energy bound as the partial differential equationdoes in the L2 norm, plus an added dissipation that depends on how much the approximatesolution fails to satisfy the Rankine–Hugoniot jump.
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| 10. |
- Wintermeyer, Niklas, et al.
(författare)
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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
- 2017
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 340, s. 200-242
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Tidskriftsartikel (refereegranskat)abstract
- We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear 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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