×

Generalised summation-by-parts operators and variable coefficients. (English) Zbl 1395.65054

Summary: High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework. However, if generalised SBP operators are used, the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely – interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain. Here, new formulations are proposed, allowing the creation of discretisations using general SBP operators that are both conservative and stable. Thus, several shortcomings that might be attributed to generalised SBP operators are overcome (cf. [J. Nordström and A. A. Ruggiu, ibid. 344, 451–464 (2017; Zbl 1380.65172); J. Manzanero et al., “Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods”, J. Sci. Comput. 75, No. 3, 1262–1281 (2017; doi:10.1007/s10915-017-0585-6)]).

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations

Citations:

Zbl 1380.65172

Software:

PyFR
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Kreiss, H.-O.; Oliger, J., Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24, 199-215, (1972)
[2] Kreiss, H.-O.; Wu, L., On the stability definition of difference approximations for the initial boundary value problem, Appl. Numer. Math., 12, 213-227, (1993) · Zbl 0782.65119
[3] Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., 110, 47-67, (1994) · Zbl 0792.65011
[4] Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments for compact high-order finite-difference schemes, J. Comput. Phys., 108, 272-295, (1993) · Zbl 0791.76052
[5] Fernández, D. C.D. R.; Hicken, J. E.; Zingg, D. W., Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. Fluids, 95, 171-196, (2014) · Zbl 1390.65064
[6] Svärd, M.; Nordström, J., Review of summation-by-parts schemes for initial-boundary-value problems, J. Comput. Phys., 268, 17-38, (2014) · Zbl 1349.65336
[7] Nordström, J.; Eliasson, P., New developments for increased performance of the SBP-SAT finite difference technique, (Kroll, N.; Hirsch, C.; Bassi, F.; Johnston, C.; Hillewaert, K., IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach, (2015), Springer), 467-488
[8] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time-dependent problems and difference methods, vol. 123, (2013), John Wiley & Sons · Zbl 1275.65048
[9] Nordström, J.; Björck, M., Finite volume approximations and strict stability for hyperbolic problems, Appl. Numer. Math., 38, 237-255, (2001) · Zbl 0985.65103
[10] Nordström, J.; Forsberg, K.; Adamsson, C.; Eliasson, P., Finite volume methods, unstructured meshes and strict stability for hyperbolic problems, Appl. Numer. Math., 45, 453-473, (2003) · Zbl 1019.65066
[11] 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, A1233-A1253, (2013) · Zbl 1275.65065
[12] Gassner, G. J., A kinetic energy preserving nodal discontinuous Galerkin spectral element method, Int. J. Numer. Methods Fluids, 76, 28-50, (2014) · Zbl 1455.76142
[13] Kopriva, D. A.; Gassner, G. J., An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems, SIAM J. Sci. Comput., 36, A2076-A2099, (2014) · Zbl 1303.65086
[14] 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., 272, 291-308, (2016) · Zbl 1410.65393
[15] Ortleb, S., A kinetic energy preserving DG scheme based on Gauss-Legendre points, J. Sci. Comput., 1-34, (2016)
[16] Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, (18th AIAA Computational Fluid Dynamics Conference, (2007), American Institute of Aeronautics and Astronautics)
[17] Wang, Z. J.; Gao, H., A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys., 228, 8161-8186, (2009) · Zbl 1173.65343
[18] Vincent, P. E.; Castonguay, P.; Jameson, A., A new class of high-order energy stable flux reconstruction schemes, J. Sci. Comput., 47, 50-72, (2011) · Zbl 1433.76094
[19] Huynh, H. T.; Wang, Z. J.; Vincent, P. E., High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids, Comput. Fluids, 98, 209-220, (2014) · Zbl 1390.65123
[20] Witherden, F. D.; Farrington, A. M.; Vincent, P. E., Pyfr: an open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach, Comput. Phys. Commun., 185, 3028-3040, (2014) · Zbl 1348.65005
[21] Vincent, P. E.; Farrington, A. M.; Witherden, F. D.; Jameson, A., An extended range of stable-symmetric-conservative flux reconstruction correction functions, Comput. Methods Appl. Mech. Eng., 296, 248-272, (2015) · Zbl 1423.76282
[22] Ranocha, H.; Öffner, P.; Sonar, T., Summation-by-parts operators for correction procedure via reconstruction, J. Comput. Phys., 311, 299-328, (2016) · Zbl 1349.65524
[23] Fernández, D. C.D. R.; Boom, P. D.; Zingg, D. W., A generalized framework for nodal first derivative summation-by-parts operators, J. Comput. Phys., 266, 214-239, (2014) · Zbl 1311.65002
[24] Hicken, J. E.; Fernández, D. C.D. R.; Zingg, D. W., Multidimensional summation-by-parts operators: general theory and application to simplex elements, SIAM J. Sci. Comput., 38, A1935-A1958, (2016) · Zbl 1382.65355
[25] Tadmor, E., The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comput., 49, 91-103, (1987) · Zbl 0641.65068
[26] Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 12, 451-512, (2003) · Zbl 1046.65078
[27] 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
[28] 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, B835-B867, (2014) · Zbl 1457.65140
[29] 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, 132-138, (2015) · Zbl 1349.76250
[30] Yamaleev, N. K.; Carpenter, M. H., A family of fourth-order entropy stable nonoscillatory spectral collocation schemes for the 1-d Navier-Stokes equations, J. Comput. Phys., 331, 90-107, (2017) · Zbl 1378.76079
[31] 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, 39-66, (2016) · Zbl 1422.65280
[32] 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., 340, 200-242, (2017) · Zbl 1380.65291
[33] Ranocha, H., Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods, GEM Int. J. Geomath., 8, 85-133, (2017) · Zbl 1432.65156
[34] Ranocha, H., Comparison of some entropy conservative numerical fluxes for the Euler equations, J. Sci. Comput., (2017), in press
[35] Nordström, J., Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation, J. Sci. Comput., 29, 375-404, (2006) · Zbl 1109.65076
[36] Fisher, T. C.; Carpenter, M. H.; Nordström, J.; Yamaleev, N. K.; Swanson, C., Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions, J. Comput. Phys., 234, 353-375, (2013) · Zbl 1284.65102
[37] Svärd, M., On coordinate transformations for summation-by-parts operators, J. Sci. Comput., 20, 29-42, (2004) · Zbl 1057.65054
[38] Nordström, J.; Ruggiu, A. A., On conservation and stability properties for summation-by-parts schemes, J. Comput. Phys., 344, 451-464, (2017) · Zbl 1380.65172
[39] Manzanero, J.; Rubio, G.; Ferrer, E.; Valero, E.; Kopriva, D. A., Insights on aliasing driven instabilities for advection equations with application to Gauss-lobatto discontinuous Galerkin methods, J. Sci. Comput., (2017)
[40] Ranocha, H., SBP operators for CPR methods, (2016), TU Braunschweig, Master’s thesis
[41] Ranocha, H.; Öffner, P.; Sonar, T., Extended skew-symmetric form for summation-by-parts operators and varying Jacobians, J. Comput. Phys., 342, 13-28, (2017) · Zbl 1380.65318
[42] 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, 136-155, (2010) · Zbl 1203.65199
[43] Hicken, J. E.; Zingg, D. W., Summation-by-parts operators and high-order quadrature, J. Comput. Appl. Math., 237, 111-125, (2013) · Zbl 1263.65025
[44] Olsson, P., Summation by parts, projections, and stability. I, Math. Comput., 64, 1035-1065, (1995) · Zbl 0828.65111
[45] Olsson, P., Summation by parts, projections, and stability. II, Math. Comput., 64, 1473-1493, (1995) · Zbl 0848.65064
[46] Mattsson, K., Boundary procedures for summation-by-parts operators, J. Sci. Comput., 18, 133-153, (2003) · Zbl 1024.76031
[47] LeVeque, R. J., Numerical methods for conservation laws, (1992), Birkhäuser Verlag P.O. Box 133, CH-4010 Basel, Switzerland · Zbl 0847.65053
[48] LeVeque, R. J., Finite volume methods for hyperbolic problems, vol. 31, (2002), Cambridge University Press · Zbl 1010.65040
[49] Bressan, A., Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem, (2000), Oxford University Press · Zbl 0997.35002
[50] Kopriva, D. A., Implementing spectral methods for partial differential equations: algorithms for scientists and engineers, (2009), Springer Science & Business Media · Zbl 1172.65001
[51] Ketcheson, D. I., Highly efficient strong stability-preserving Runge-Kutta methods with low-storage implementations, SIAM J. Sci. Comput., 30, 2113-2136, (2008) · Zbl 1168.65382
[52] Nordström, J., Error bounded schemes for time-dependent hyperbolic problems, SIAM J. Sci. Comput., 30, 46-59, (2007) · Zbl 1171.35308
[53] Kopriva, D. A.; Nordström, J.; Gassner, G. J., Error boundedness of discontinuous Galerkin spectral element approximations of hyperbolic problems, J. Sci. Comput., 72, 314-330, (2017) · Zbl 1371.65097
[54] Gassner, G. J.; Kopriva, D. A., A comparison of the dispersion and dissipation errors of Gauss and Gauss-lobatto discontinuous Galerkin spectral element methods, SIAM J. Sci. Comput., 33, 2560-2579, (2011) · Zbl 1255.65089
[55] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods: fundamentals in single domains, (2006), Springer Berlin, Heidelberg · Zbl 1093.76002
[56] Warburton, T.; Hagstrom, T., Taming the CFL number for discontinuous Galerkin methods on structured meshes, SIAM J. Numer. Anal., 46, 3151-3180, (2008) · Zbl 1181.35010
[57] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261, (2001) · Zbl 1065.76135
[58] LeFloch, P. G.; Mercier, J.-M.; Rohde, C., Fully discrete, entropy conservative schemes of arbitrary order, SIAM J. Numer. Anal., 40, 1968-1992, (2002) · Zbl 1033.65073
[59] R.D. Richtmyer, K.W. Morton, Difference methods for boundary-value problems, 1967.; R.D. Richtmyer, K.W. Morton, Difference methods for boundary-value problems, 1967. · Zbl 0155.47502
[60] von Neumann, J.; Richtmyer, R. D., A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21, 232-237, (1950) · Zbl 0037.12002
[61] Tadmor, E., Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26, 30-44, (1989) · Zbl 0667.65079
[62] Grahs, T.; Sonar, T., Entropy-controlled artificial anisotropic diffusion for the numerical solution of conservation laws based on algorithms from image processing, J. Vis. Commun. Image Represent., 13, 176-194, (2002)
[63] Grahs, T.; Meister, A.; Sonar, T., Image processing for numerical approximations of conservation laws: nonlinear anisotropic artificial dissipation, SIAM J. Sci. Comput., 23, 1439-1455, (2002) · Zbl 1006.65090
[64] Breuß, M.; Bürgel, A.; Brox, T.; Sonar, T.; Weickert, J., Numerical aspects of TV flow, Numer. Algorithms, 41, 79-101, (2006) · Zbl 1116.65099
[65] Bianchini, S.; Bressan, A., Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 223-342, (2005) · Zbl 1082.35095
[66] Mattsson, K.; Svärd, M.; Nordström, J., Stable and accurate artificial dissipation, J. Sci. Comput., 21, 57-79, (2004) · Zbl 1085.76050
[67] Persson, P.-O.; Peraire, J., Sub-cell shock capturing for discontinuous Galerkin methods, (44th AIAA Aerospace Sciences Meeting and Exhibit, (2006), American Institute of Aeronautics and Astronautics)
[68] Barter, G. E.; Darmofal, D. L., Shock capturing with PDE-based artificial viscosity for DGFEM: part I. formulation, J. Comput. Phys., 229, 1810-1827, (2010) · Zbl 1329.76153
[69] Guermond, J.-L.; Pasquetti, R.; Popov, B., From suitable weak solutions to entropy viscosity, (Quality and Reliability of Large-Eddy Simulations II, (2011), Springer), 373-390 · Zbl 1303.76007
[70] Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248-4267, (2011) · Zbl 1220.65134
[71] Grahs, T.; Sonar, T., Data analysis and entropy steered discrete filtering for the numerical treatment of conservation laws, Int. J. Numer. Methods Fluids, 40, 353-359, (2002) · Zbl 1035.76035
[72] Bürgel, A.; Grahs, T.; Sonar, T., From continuous recovery to discrete filtering in numerical approximations of conservation laws, Appl. Numer. Math., 42, 47-60, (2002) · Zbl 0998.65094
[73] Hesthaven, J.; Kirby, R., Filtering in Legendre spectral methods, Math. Comput., 77, 1425-1452, (2008) · Zbl 1195.65138
[74] Meister, A.; Ortleb, S.; Sonar, T., Application of spectral filtering to discontinuous Galerkin methods on triangulations, Numer. Methods Partial Differ. Equ., 28, 1840-1868, (2012) · Zbl 1251.65141
[75] Meister, A.; Ortleb, S.; Sonar, T.; Wirz, M., A comparison of the discontinuous-Galerkin- and spectral-difference-method on triangulations using PKD polynomials, J. Comput. Phys., 231, 7722-7729, (2012) · Zbl 1257.65054
[76] Meister, A.; Ortleb, S.; Sonar, T.; Wirz, M., An extended discontinuous Galerkin and spectral difference method with modal filtering, J. Appl. Math. Mech., 93, 459-464, (2013) · Zbl 1275.65062
[77] Huerta, A.; Casoni, E.; Peraire, J., A simple shock-capturing technique for high-order discontinuous Galerkin methods, Int. J. Numer. Methods Fluids, 69, 1614-1632, (2012) · Zbl 1253.76058
[78] Dumbser, M.; Zanotti, O.; Loubère, R.; Diot, S., A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J. Comput. Phys., 278, 47-75, (2014) · Zbl 1349.65448
[79] Sonntag, M.; Munz, C.-D., Shock capturing for discontinuous Galerkin methods using finite volume subcells, (Fuhrmann, J.; Ohlberger, M.; Rohde, C., Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Springer Proceedings in Mathematics & Statistics, vol. 78, (2014), Springer International Publishing), 945-953 · Zbl 1426.76429
[80] Meister, A.; Ortleb, S., A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions, Appl. Math. Comput., 272, 259-273, (2016) · Zbl 1410.76250
[81] Sonntag, M.; Munz, C.-D., Efficient parallelization of a shock capturing for discontinuous Galerkin methods using finite volume sub-cells, J. Sci. Comput., 1-28, (2016)
[82] Fjordholm, U. S.; Mishra, S.; Tadmor, E., Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50, 544-573, (2012) · Zbl 1252.65150
[83] Fjordholm, U. S., High-order accurate entropy stable numerical schemes for hyperbolic conservation laws, (2013), ETH Zürich, Ph.D. thesis
[84] Fisher, T. C., High-order \(L^2\) stable multi-domain finite difference method for compressible flows, (2012), Purdue University, Ph.D. thesis
[85] Kopriva, D. A., Metric identities and the discontinuous spectral element method on curvilinear meshes, J. Sci. Comput., 26, 301-327, (2006) · Zbl 1178.76269
[86] LeFloch, P. G., Hyperbolic systems of conservation laws: the theory of classical and nonclassical shock waves, (2002), Springer Science & Business Media · Zbl 1019.35001
[87] Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21, 217-235, (1984) · Zbl 0592.65069
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.