# zbMATH — the first resource for mathematics

Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. (English) Zbl 1422.65280
Summary: In [ibid. 252, 518–557 (2013; Zbl 1349.65293)], T. C. Fisher and M. H. Carpenter 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 subcell 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 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.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q31 Euler equations
Full Text:
##### References:
 [1] Blaisdell, G. A.; Spyropoulos, E. T.; Qin, J. H., The effect of the formulation of nonlinear terms on aliasing errors in spectral methods, Appl. Numer. Math., 21, 3, 207-219, (1996) · Zbl 0858.76060 [2] Carpenter, M.; Fisher, T.; Nielsen, E.; Frankel, S., Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces, SIAM J. Sci. Comput., 36, 5, B835-B867, (2014) · Zbl 1457.65140 [3] Carpenter, M.; Kennedy, C., Fourth-order 2N-storage Runge-Kutta schemes, (1994), NASA Langley Research Center, NASA TM 109111 [4] Chandrashekar, P., Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys., 14, 5, 1252-1286, (2013) · Zbl 1373.76121 [5] Cockburn, B.; Hou, S.; Shu, C. W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 545-581, (1990) · Zbl 0695.65066 [6] Cockburn, B.; Lin, S. Y.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093 [7] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083 [8] Cockburn, B.; Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059 [9] Cockburn, B.; Shu, C. W., The Runge-Kutta local projection $$p^1$$-discontinuous Galerkin method for scalar conservation laws, M^{2}AN, 25, 337-361, (1991) · Zbl 0732.65094 [10] Ducros, F.; Laporte, F.; Soulères, T.; Guinot, V.; Moinat, P.; Caruelle, B., High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows, J. Comput. Phys., 161, 114-139, (2000) · Zbl 0972.76066 [11] Fisher, T. C., High-order $$L_2$$ stable multi-domain finite difference method for compressible flows, (2012), Purdue University, PhD thesis [12] Fisher, T. C.; Carpenter, M. H., High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains, J. Comput. Phys., 252, 518-557, (2013) · Zbl 1349.65293 [13] Gassner, G. J., A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, 3, A1233-A1253, (2013) · Zbl 1275.65065 [14] Gassner, G. J., A kinetic energy preserving nodal discontinuous Galerkin spectral element method, Int. J. Numer. Methods Fluids, 76, 1, 28-50, (2014) [15] Gassner, G. J.; A. D., Beck, On the accuracy of high-order discretizations for underresolved turbulence simulations, Theor. Comput. Fluid Dyn., (2012) [16] Gassner, G. J.; Winters, A. R.; Kopriva, D. A., A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations, Appl. Math. Comput., (2015) [17] Hiltebrand, A.; Mishra, S., Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numer. Math., 126, 1, 103-151, (2014) · Zbl 1303.65083 [18] Hindenlang, F.; Gassner, G.; Altmann, C.; Beck, A.; Staudenmaier, M.; Munz, C.-D., Explicit discontinuous Galerkin methods for unsteady problems, Comput. Fluids, 61, 86-93, (2012) · Zbl 1365.76117 [19] Ismail, F.; Roe, P. L., Affordable, entropy-consistent Euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 15, 5410-5436, (2009) · Zbl 1280.76015 [20] Jameson, A., Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes, J. Sci. Comput., 34, 3, 188-208, (2008) · Zbl 1133.76031 [21] Jiang, G.; Shu, C.-W., On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput., 62, 206, 531-538, (1994) · Zbl 0801.65098 [22] Kennedy, C. A.; Gruber, A., Reduced aliasing formulations of the convective terms within the Navier-Stokes equations for a compressible fluid, J. Comput. Phys., 227, 1676-1700, (2008) · Zbl 1290.76135 [23] Kirby, R. M.; Karniadakis, G. E., De-aliasing on non-uniform grids: algorithms and applications, J. Comput. Phys., 191, 249-264, (2003) · Zbl 1161.76534 [24] Kopriva, D. A., Metric identities and the discontinuous spectral element method on curvilinear meshes, J. Sci. Comput., 26, 3, 301-327, (March 2006) [25] Kopriva, D. A., Implementing spectral methods for partial differential equations: algorithms for scientists and engineers, (2009), Springer Publishing Company, Incorporated · Zbl 1172.65001 [26] Kopriva, D. A.; Gassner, G. J., On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods, J. Sci. Comput., 44, 2, 136-155, (2010) · Zbl 1203.65199 [27] Kopriva, D. A.; Gassner, G. J., An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems, SIAM J. Sci. Comput., 36, 4, 2076-2099, (2014) [28] Kravchenko, A. G.; Moin, P., On the effect of numerical errors in large eddy simulations of turbulent flows, J. Comput. Phys., 131, 2, 310-322, (1997) · Zbl 0872.76074 [29] Larsson, J.; Lele, S. K.; Moin, P., Effect of numerical dissipation on the predicted spectra for compressible turbulence, Annual Research Briefs, 47-57, (2007) [30] Mengaldo, G.; De Grazia, D.; Moxey, D.; Vincent, P. E.; Sherwin, S. J., Dealiasing techniques for high-order spectral element methods on regular and irregular grids, J. Comput. Phys., 299, 56-81, (2015) · Zbl 1352.65396 [31] Morinishi, Y., Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows, J. Comput. Phys., 229, 2, 276-300, (2010) · Zbl 1375.76113 [32] Moura, R. C.; Sherwin, S. J.; Peiro, J., On DG-based iles approaches at very high Reynolds numbers, (2015), Report, Research Gate [33] Parsani, M.; Carpenter, M. H.; Nielsen, E. J., Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 290(C), 132-138, (June 2015) [34] Parsani, M.; Carpenter, M. H.; Nielsen, E. J., Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 292, 88-113, (2015) · Zbl 1349.76639 [35] Pirozzoli, S., Generalized conservative approximations of split convective derivative operators, J. Comput. Phys., 229, 19, 7180-7190, (2010) · Zbl 1426.76485 [36] Pirozzoli, S., Numerical methods for high-speed flows, Annu. Rev. Fluid Mech., 43, 163-194, (2011) · Zbl 1299.76103 [37] Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., 110, 1, 47-67, (1994) · Zbl 0792.65011 [38] Svärd, M.; Özcan, Hatice, Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions, J. Sci. Comput., 58, 1, 61-89, (2014) · Zbl 1290.65084 [39] Wintermeyer, N.; Winters, A. R.; Gassner, G. J.; Kopriva, D. A., An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry, J. Comput. Phys., (2015), submitted for publication · Zbl 1380.65291 [40] Zang, T. A., On the rotation and skew-symmetric forms for incompressible flow simulations, Appl. Numer. Math., 7, 27-40, (1991) · Zbl 0708.76071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.