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Journal of Scientific Computing

Erschienen in:

08.11.2017

Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

verfasst von: Jianfeng Yan, Jared Crean, Jason E. Hicken

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2018

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Abstract

This work focuses on simultaneous approximation terms (SATs) for multidimensional summation-by-parts (SBP) discretizations of linear second-order partial differential equations with variable coefficients. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties for general operators, including those operators that do not have nodes on their boundary or do not correspond with a collocation discontinuous Galerkin method. Based on these conditions, we generalize the modified scheme of Bassi and Rebay and the symmetric interior penalty Galerkin method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.

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Equivalently, we can consider the generalized eigenvalue problem \(\mathsf {A} \varvec{v}_{i,j} = \mu _{i,j}\mathsf {H} \varvec{v}_{i,j}\); see, for example, [38, Chapter 8]

Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252(1), 518–557 (2013)MathSciNetCrossRefMATH

Fisher, T.C.: High-order L2 stable multi-domain finite difference method for compressible flows. Ph.D. thesis, Purdue University (2012)

Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York (1974)

Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110(1), 47–67 (1994)MathSciNetCrossRefMATH

Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier–Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835–B867 (2014)MathSciNetCrossRefMATH

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(C), 88–113 (2015)MathSciNetCrossRefMATH

Gassner, G .J., Winters, A .R., Kopriva, D .A.: Split form nodal discontinuous Galerkin Schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327(C), 39–66 (2016)MathSciNetCrossRefMATH

Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017)MathSciNetCrossRefMATH

Crean, J., Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. J. Comput. Phys. (in revision) (2017)

10.

Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Multi-dimensional summation-by-parts operators: general theory and application to simplex elements. SIAM J. Sci. Comput. 38(4), A1935–A1958 (2016)CrossRefMATH

11.

Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148(2), 341–365 (1999)MathSciNetCrossRefMATH

12.

Carpenter, M.H., Nordström, J., Gottlieb, D.: Revisiting and extending interface penalties for multi-domain summation-by-parts operators. J. Sci. Comput. 45(1), 118–150 (2010)MathSciNetCrossRefMATH

13.

Mattsson, K., Carpenter, M.H.: Stable and accurate interpolation operators for high-order multiblock finite difference methods. SIAM J. Sci. Comput. 32(4), 2298–2320 (2010)MathSciNetCrossRefMATH

14.

Gong, J., Nordström, J.: Interface procedures for finite difference approximations of the advection–diffusion equation. J. Comput. Appl. Math. 236(5), 602–620 (2011)MathSciNetCrossRefMATH

15.

Del Rey Fernández, D.C., Zingg, D .W.: Generalized summation-by-parts operators for the second derivative with a variable coefficient. SIAM J. Sci. Comput 37(6), A2840–A2864 (2015)CrossRefMATH

16.

Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRefMATH

17.

Warburton, T., Hesthaven, J.: On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192(25), 2765–2773 (2003)MathSciNetCrossRefMATH

18.

Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3), 311–341 (1999)MathSciNetCrossRefMATH

19.

Bassi, F., Crivellini, A., Rebay, S., Savini, M.: Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k turbulence model equations. Comput. Fluids 34(45), 507–540 (2005)CrossRefMATH

20.

Douglas, J., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods computing methods in applied sciences. In: Glowinski, R., Lions, J.L. (ed), Computing Methods in Applied Sciences, Volume 58 of Lecture Notes in Physics, Chap. 6, pp. 207–216. Springer, Berlin (1976)

21.

Hartmann, R.: Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM J. Numer. Anal. 45(6), 2671–2696 (2007)MathSciNetCrossRefMATH

22.

Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Simultaneous approximation terms for multi-dimensional summation-by-parts operators. In: 46th AIAA Fluid Dynamics Conference, Washington, DC, AIAA-2016-3971 (June 2016)

23.

Del Rey Fernández, D.C., Hicken, J.E., Zingg, D.W.: Simultaneous approximation terms for multi-dimensional summation-by-parts operators. J. Sci. Comput. 1–28 (2017). https://doi.org/10.1007/s10915-017-0523-7

24.

Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51(3), 650–682 (2012)MathSciNetCrossRefMATH

25.

Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163–1227 (2014)MathSciNetCrossRefMATH

26.

Babuška, I., Miller, A.: The post-processing approach in the finite element methodpart 1: calculation of displacements, stresses and other higher derivatives of the displacements. Int. J. Numer. Methods Eng. 20(6), 1085–1109 (1984)CrossRefMATH

27.

Babuška, I., Miller, A.: The post-processing approach in the finite element methodPart 2: the calculation of stress intensity factors. Int. J. Numer. Methods Eng. 20(6), 1111–1129 (1984)CrossRefMATH

28.

Pierce, N.A., Giles, M.B.: Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev. 42(2), 247–264 (2000)MathSciNetCrossRefMATH

29.

Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002)MathSciNetCrossRefMATH

30.

Lu, J.C.: An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2005)

31.

Fidkowski, K.J., Darmofal, D.L.: Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA J. 49(4), 673–694 (2011)CrossRef

32.

Hicken, J.E., Zingg, D.W.: Superconvergent functional estimates from summation-by-parts finite-difference discretizations. SIAM J. Sci. Comput. 33(2), 893–922 (2011)MathSciNetCrossRefMATH

33.

Hicken, J.E., Zingg, D.W.: Dual consistency and functional accuracy: a finite-difference perspective. J. Comput. Phys. 256, 161–182 (2014)MathSciNetCrossRefMATH

34.

Antonietti, P.F., Buffa, A., Perugia, I.: Discontinuous Galerkin approximation of the Laplace eigenproblem. Comput. Methods Appl. Mech. Eng. 195(25), 3483–3503 (2006)MathSciNetCrossRefMATH

35.

Shahbazi, K., Mavriplis, D.J., Burgess, N.K.: Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 228(21), 7917–7940 (2009)MathSciNetCrossRefMATH

36.

Peraire, J., Persson, P.-O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30(4), 1806–1824 (2008)MathSciNetCrossRefMATH

37.

Shahbazi, K.: An explicit expression for the penalty parameter of the interior penalty method. J. Comput. Phys. 205(2), 401–407 (2005)CrossRefMATH

38.

Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)CrossRefMATH

39.

Kirby, R.M., Karniadakis, G.E.: Selecting the numerical flux in discontinuous Galerkin methods for diffusion problems. J. Sci. Comput. 22–23(1–3), 385–411 (2005)MathSciNetCrossRefMATH

Titel: Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations
verfasst von: Jianfeng Yan
Jared Crean
Jason E. Hicken
Publikationsdatum: 08.11.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0591-8

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