High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. (English) Zbl 1349.76324
Summary: This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by the second and the third author [Contin. Mech. Thermodyn. 28, No. 1–2, 85–104 (2016; Zbl 1348.76046)], further denoted as HPR model. In that framework, the viscous stresses are computed from the so-called distortion tensor A, which is one of the primary state variables in the proposed first order system. A very important key feature of the HPR model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids. Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of \(A\). Also heat conduction is included via a first order hyperbolic system for the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the first and the second principle of thermodynamics. It is also fundamentally different from first order Maxwell-Cattaneo-type relaxation models based on extended irreversible thermodynamics. The HPR model represents therefore a novel and unified description of continuum mechanics, which applies at the same time to fluid mechanics and solid mechanics. In this paper, the direct connection between the HPR model and the classical hyperbolic-parabolic Navier-Stokes-Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit. From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER-WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemes to the HPR model in the stiff relaxation limit, and compare the numerical results with exact or numerical reference solutions obtained for the Euler and Navier-Stokes equations. Numerical convergence results are also provided. To show the universality of the HPR model, the paper is rounded-off with an application to wave propagation in elastic solids, for which one only needs to switch off the strain relaxation source term in the governing PDE system. We provide various examples showing that for the purpose of flow visualization, the distortion tensor \(A\) seems to be particularly useful.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 74S10 | Finite volume methods applied to problems in solid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65M08 | Finite volume 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 |
Keywords:
ADER-WENO finite volume schemes; arbitrary high-order discontinuous Galerkin schemes; path-conservative methods and stiff source terms; unified first order hyperbolic formulation of nonlinear continuum mechanics; fluid mechanics and solid mechanics; viscous compressible fluids and elastic solidsCitations:
Zbl 1348.76046References:
| [1] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779 (2002) · Zbl 1008.65080 |
| [2] | Babucke, A.; Kloker, M.; Rist, U., DNS of a plane mixing layer for the investigation of sound generation mechanisms, Comput. Fluids, 37, 360-368 (2008) · Zbl 1237.76163 |
| [3] | Balsara, D.; Shu, C. W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452 (2000) · Zbl 0961.65078 |
| [4] | Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228, 2480-2516 (2009) · Zbl 1275.76169 |
| [5] | Balsara, D. S.; Dumbser, M., Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys., 299, 687-715 (2015) · Zbl 1351.76092 |
| [6] | Balsara, D. S.; Meyer, C.; Dumbser, M.; Du, H.; Xu, Z., Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - speed comparisons with Runge-Kutta methods, J. Comput. Phys., 235, 934-969 (2013) · Zbl 1291.76237 |
| [7] | Barton, P. T.; Deiterding, R.; Meiron, D.; Pullin, D., Eulerian adaptive finite-difference method for high-velocity impact and penetration problems, J. Comput. Phys., 240, 76-99 (2013) |
| [8] | Barton, P. T.; Drikakis, D.; Romenski, E.; Titarev, V. A., Exact and approximate solutions of Riemann problems in non-linear elasticity, J. Comput. Phys., 228, 18, 7046-7068 (2009) · Zbl 1172.74032 |
| [9] | Barton, P. T.; Drikakis, D.; Romenski, E. I., An Eulerian finite-volume scheme for large elastoplastic deformations in solids, Int. J. Numer. Methods Eng., 81, 4, 453-484 (2010) · Zbl 1183.74331 |
| [10] | Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279 (1997) · Zbl 0871.76040 |
| [11] | Baumann, C. E.; Oden, J. T., A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Eng., 175, 3-4, 311-341 (1999) · Zbl 0924.76051 |
| [12] | Baumann, C. E.; Oden, J. T., A discontinuous hp finite element method for the Euler and Navier-Stokes equations, Int. J. Numer. Methods Fluids, 31, 1, 79-95 (1999) · Zbl 0985.76048 |
| [13] | Becker, R., Stosswelle und Detonation, Physik, 8, 321 (1923) |
| [14] | Bell, J. B.; Coletta, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257-283 (1989) · Zbl 0681.76030 |
| [15] | Ben-Artzi, M.; Li, J.; Warnecke, G., A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 218, 19-43 (2006) · Zbl 1158.76375 |
| [16] | Besseling, J. F., A thermodynamic approach to rheology, (Parkus, H.; Sedov, L. I., Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids, IUTAM Symposia (1968), Springer: Springer Vienna), 16-53 · Zbl 0264.76005 |
| [17] | Blasius, H., Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys., 56, 1-37 (1908) · JFM 39.0803.02 |
| [18] | Bolmatov, D.; Brazhkin, V. V.; Trachenko, K., Thermodynamic behaviour of supercritical matter, Nat. Commun., 4 (2013) |
| [19] | Bolmatov, D.; Zhernenkov, M.; Zav’yalov, D.; Stoupin, S.; Cai, Y. Q.; Cunsolo, A., Revealing the mechanism of the viscous-to-elastic crossover in liquids, J. Phys. Chem. Lett., 6, 15, 3048-3053 (2015) |
| [20] | Bonnet, A.; Luneau, J., Aérodynamique. Théories de la dynamique des fluides (1989), Cepadues Editions: Cepadues Editions Toulouse · Zbl 0744.76008 |
| [21] | Boscheri, W.; Dumbser, M., A direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and nonconservative hyperbolic systems in 3D, J. Comput. Phys., 275, 484-523 (2014) · Zbl 1349.76310 |
| [22] | Brachet, M. E.; Meiron, D. I.; Orszag, S. A.; Nickel, B. G.; Morf, R. H.; Frisch, U., Small-scale structure of the Taylor-Green vortex, J. Fluid Mech., 130, 411-452 (1983) · Zbl 0517.76033 |
| [23] | Brazhkin, V. V.; Fomin, Yu. D.; Lyapin, A. G.; Ryzhov, V. N.; Trachenko, K., Two liquid states of matter: a dynamic line on a phase diagram, Phys. Rev. E, 85, 3, Article 031203 pp. (2012) |
| [24] | Castro, C. C.; Toro, E. F., Solvers for the high-order Riemann problem for hyperbolic balance laws, J. Comput. Phys., 227, 2481-2513 (2008) · Zbl 1148.65066 |
| [25] | Castro, M. J.; Gallardo, J. M.; Marquina, A., Approximate Osher-Solomon schemes for hyperbolic systems, Appl. Math. Comput., 272, 347-368 (2016) · Zbl 1410.76325 |
| [26] | Castro, M. J.; Gallardo, J. M.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comput., 75, 1103-1134 (2006) · Zbl 1096.65082 |
| [27] | Castro, M. J.; LeFloch, P. G.; Muñoz-Ruiz, M. L.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys., 227, 8107-8129 (2008) · Zbl 1176.76084 |
| [28] | Cattaneo, C., Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena, 3 (1948) · Zbl 0035.26203 |
| [29] | Cesenek, J.; Feistauer, M.; Horacek, J.; Kucera, V.; Prokopova, J., Simulation of compressible viscous flow in time-dependent domains, Appl. Math. Comput., 219, 7139-7150 (2013) · Zbl 1426.76233 |
| [30] | Chisolm, E. D.; Wallace, D. C., Dynamics of monatomic liquids, J. Phys. Condens. Matter, 13, 37, Article R739 pp. (2001) |
| [31] | Cockburn, B.; Shu, C. W., The local discontinuous Galerkin method for time-dependent convection diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118 |
| [32] | Cockburn, B.; Shu, C. W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261 (2001) · Zbl 1065.76135 |
| [33] | Colonius, T.; Lele, S. K.; Moin, P., Sound generation in a mixing layer, J. Fluid Mech., 330, 375-409 (1997) · Zbl 0901.76075 |
| [34] | Courant, R.; Friedrichs, K.; Lewy, H., Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100, 1, 32-74 (1928) · JFM 54.0486.01 |
| [35] | Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics (2005), Springer-Verlag: Springer-Verlag Berlin · Zbl 1078.35001 |
| [36] | Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390 (1991) · Zbl 0742.76059 |
| [37] | Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput. Fluids, 39, 60-76 (2010) · Zbl 1242.76161 |
| [38] | Dumbser, M.; Balsara, D.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes, J. Comput. Phys., 227, 8209-8253 (2008) · Zbl 1147.65075 |
| [39] | Dumbser, M.; Balsara, D. S., A new, efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems, J. Comput. Phys., 304, 275-319 (2016) · Zbl 1349.76603 |
| [40] | Dumbser, M.; Boscheri, W., High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows, Comput. Fluids, 86, 405-432 (2013) · Zbl 1290.76081 |
| [41] | Dumbser, M.; Castro, M.; Parés, C.; Toro, E. F., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Comput. Fluids, 38, 1731-1748 (2009) · Zbl 1177.76222 |
| [42] | Dumbser, M.; Casulli, V., A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier-Stokes equations with general equation of state, Appl. Math. Comput., 272, 479-497 (2016) · Zbl 1410.76220 |
| [43] | Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227, 3971-4001 (2008) · Zbl 1142.65070 |
| [44] | Dumbser, M.; Facchini, M., A local space-time discontinuous Galerkin method for Boussinesq-type equations, Appl. Math. Comput., 272, 336-346 (2016) · Zbl 1410.76167 |
| [45] | Dumbser, M.; Hidalgo, A.; Castro, M.; Parés, C.; Toro, E. F., FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput. Methods Appl. Mech. Eng., 199, 625-647 (2010) · Zbl 1227.76043 |
| [46] | Dumbser, M.; Hidalgo, A.; Zanotti, O., High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems, Comput. Methods Appl. Mech. Eng., 268, 359-387 (2014) · Zbl 1295.65088 |
| [47] | Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 693-723 (2007) · Zbl 1110.65077 |
| [48] | Dumbser, M.; Käser, M.; Titarev, V. A.; Toro, E. F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226, 204-243 (2007) · Zbl 1124.65074 |
| [49] | Dumbser, M.; Munz, C. D., Building blocks for arbitrary high order discontinuous Galerkin schemes, J. Sci. Comput., 27, 215-230 (2006) · Zbl 1115.65100 |
| [50] | Dumbser, M.; Toro, E. F., On universal Osher-type schemes for general nonlinear hyperbolic conservation laws, Commun. Comput. Phys., 10, 635-671 (2011) · Zbl 1373.76125 |
| [51] | Dumbser, M.; Toro, E. F., A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems, J. Sci. Comput., 48, 70-88 (2011) · Zbl 1220.65110 |
| [52] | Dumbser, M.; Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, J. Comput. Phys., 228, 6991-7006 (2009) · Zbl 1261.76028 |
| [53] | Dumbser, M.; Zanotti, O.; Hidalgo, A.; Balsara, D. S., ADER-WENO finite volume schemes with space-time adaptive mesh refinement, J. Comput. Phys., 248, 257-286 (2013) · Zbl 1349.76325 |
| [54] | 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 (December 2014) · Zbl 1349.65448 |
| [55] | Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 294-318 (1988) · Zbl 0642.76088 |
| [56] | Einfeldt, B.; Munz, C. D.; Roe, P. L.; Sjögreen, B., On Godunov-type methods near low densities, J. Comput. Phys., 92, 273-295 (1991) · Zbl 0709.76102 |
| [57] | Favrie, N.; Gavrilyuk, S. L.; Saurel, R., Solid-fluid diffuse interface model in cases of extreme deformations, J. Comput. Phys., 228, 16, 6037-6077 (2009) · Zbl 1280.74013 |
| [58] | Favrie, N.; Gavrilyuk, S. L., Diffuse interface model for compressible fluid - compressible elastic-plastic solid interaction, J. Comput. Phys., 231, 2695-2723 (2012) · Zbl 1430.74036 |
| [59] | Le Floch, P.; Tatsien, L., A global asymptotic expansion for the solution of the generalized Riemann problem, Ann. Inst. Henri Poincaré, C Anal. Non Linéaire, 3, 321-340 (1991) · Zbl 0731.35006 |
| [60] | Frenkel, J., Kinetic Theory of Liquids (1955), Dover · Zbl 0063.01447 |
| [61] | Gassner, G.; Dumbser, M.; Hindenlang, F.; Munz, C. D., Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors, J. Comput. Phys., 230, 4232-4247 (2011) · Zbl 1220.65122 |
| [62] | Gassner, G.; Lörcher, F.; Munz, C. D., A discontinuous Galerkin scheme based on a space-time expansion II. Viscous flow equations in multi dimensions, J. Sci. Comput., 34, 260-286 (2008) · Zbl 1218.76027 |
| [63] | Gassner, G.; Lörcher, F.; Munz, C. D., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, J. Comput. Phys., 224, 1049-1063 (2007) · Zbl 1123.76040 |
| [64] | Gavrilyuk, S. L.; Favrie, N.; Saurel, R., Modelling wave dynamics of compressible elastic materials, J. Comput. Phys., 227, 2941-2969 (2008) · Zbl 1155.74020 |
| [65] | Ghia, U.; Ghia, K. N.; Shin, C. T., High-Re solutions for incompressible flow using Navier-Stokes equations and multigrid method, J. Comput. Phys., 48, 387-411 (1982) · Zbl 0511.76031 |
| [66] | Godlewski, E.; Raviart, P. A., Numerical Approximation of Hyperbolic Systems of Conservation Laws (1996), Springer · Zbl 1063.65080 |
| [67] | Godunov, Sergei Konstantinovich, The problem of a generalized solution in the theory of quasilinear equations and in gas dynamics, Russ. Math. Surv., 17, 3, 145-156 (1962) · Zbl 0107.20003 |
| [68] | Godunov, S. K., An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139, 3, 521-523 (1961) · Zbl 0125.06002 |
| [69] | Godunov, S. K., Symmetric form of the magnetohydrodynamic equation, Numer. Methods Mech. Contin. Medium, 3, 1, 26-34 (1972) |
| [70] | Godunov, S. K., Elements of Mechanics of Continuous Media (1978), Nauka, (in Russian) |
| [71] | Godunov, S. K., Equations of Mathematical Physics (1979), Nauka: Nauka Moscow, (in Russian) |
| [72] | Godunov, S. K.; Mikhailova, T. Yu.; Romenski, E. I., Systems of thermodynamically coordinated laws of conservation invariant under rotations, Sib. Math. J., 37, 4, 690-705 (1996) · Zbl 0891.73003 |
| [73] | Godunov, S. K.; Peshkov, I. M., Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium, Comput. Math. Math. Phys., 50, 8, 1409-1426 (2010) · Zbl 1224.74017 |
| [74] | Godunov, S. K.; Romenski, E. I., Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates, J. Appl. Mech. Tech. Phys., 13, 6, 868-884 (1972) |
| [75] | Godunov, S. K.; Romenski, E. I., Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media, Comput. Fluid Dyn. Rev., 95, 19-31 (1995) · Zbl 0875.73025 |
| [76] | Godunov, S. K.; Romenski, E. I., Symmetric forms of thermodynamically compatible systems of conservation laws in continuum mechanics, (ECCOMAS Conference on Numerical Methods in Engineering (1996)), 54-57 |
| [77] | Godunov, S. K.; Romenski, E. I., Elements of Mechanics of Continuous Media (1998), Nauchnaya Kniga, (in Russian) · Zbl 1053.74001 |
| [78] | Godunov, S. K.; Romenski, E. I., Elements of Continuum Mechanics and Conservation Laws (2003), Kluwer Academic/Plenum Publishers · Zbl 1031.74004 |
| [79] | Gundlach, C.; Hawke, I.; Erickson, S. J., A conservation law formulation of nonlinear elasticity in general relativity, Class. Quantum Gravity, 29, 1, Article 015005 pp. (January 2012) · Zbl 1235.83015 |
| [80] | Hartmann, R.; Houston, P., Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: method formulation, Int. J. Numer. Anal. Model., 3, 1-20 (2006) · Zbl 1129.76030 |
| [81] | Hartmann, R.; Houston, P., An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations, J. Comput. Phys., 227, 9670-9685 (2008) · Zbl 1359.76220 |
| [82] | Hidalgo, A.; Dumbser, M., ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J. Sci. Comput., 48, 173-189 (2011) · Zbl 1221.65231 |
| [83] | Hu, C.; Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127 (1999) · Zbl 0926.65090 |
| [84] | Karniadakis, G. E.; Sherwin, S. J., Spectral/hp Element Methods in CFD (1999), Oxford University Press · Zbl 0954.76001 |
| [85] | Käser, M.; Dumbser, M., An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes I: the two-dimensional isotropic case with external source terms, Geophys. J. Int., 166, 855-877 (2006) |
| [86] | Klaij, C.; Van der Vegt, J. J.W.; Van der Ven, H., Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., 217, 589-611 (2006) · Zbl 1099.76035 |
| [87] | Komatitsch, D.; Tromp, J., Introduction to the spectral-element method for 3-d seismic wave propagation, Geophys. J. Int., 139, 806-822 (1999) |
| [88] | Komatitsch, D.; Vilotte, J. P., The spectral-element method: an efficient tool to simulate the seismic response of 2d and 3d geological structures, Bull. Seismol. Soc. Am., 88, 368-392 (1998) · Zbl 0974.74583 |
| [89] | Kulikovskii, A. G.; Pogorelov, N. V.; Semenov, A. Yu., Mathematical Aspects of Numerical Solution of Hyperbolic Systems (2000), CRC Press · Zbl 0965.35001 |
| [90] | Lamb, H., On the propagation of tremors over the surface of an elastic solid, Philos. Trans. R. Soc. Lond. Ser. A, 203, 1-42 (1904) · JFM 34.0859.02 |
| [91] | Landau, L. D.; Lifshitz, E. M., Fluid Mechanics, Course of Theoretical Physics, vol. 6 (2004), Elsevier Butterworth-Heinemann: Elsevier Butterworth-Heinemann Oxford |
| [92] | Levy, D.; Shu, C. W.; Yan, J., Local discontinuous Galerkin methods for nonlinear dispersive equations, J. Comput. Phys., 196, 751-772 (2004) · Zbl 1055.65109 |
| [93] | Loubère, R.; Dumbser, M.; Diot, S., A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun. Comput. Phys., 16, 718-763 (2014) · Zbl 1373.76137 |
| [94] | Luo, H.; Luo, L.; Nourgaliev, R.; Mousseau, V. A.; Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys., 229, 6961-6978 (2010) · Zbl 1425.35138 |
| [95] | Luo, H.; Xia, Y.; Spiegel, S.; Nourgaliev, R.; Jiang, Z., A reconstructed discontinuous Galerkin method based on a hierarchical WENO reconstruction for compressible flows on tetrahedral grids, J. Comput. Phys., 236, 477-492 (2013) · Zbl 1286.65125 |
| [96] | Malyshev, A. N.; Romenski, E. I., Hyperbolic equations for heat transfer. Global solvability of the Cauchy problem, Sib. Math. J., 27, 5, 734-740 (1986) · Zbl 0654.35062 |
| [97] | Dal Maso, G.; LeFloch, P. G.; Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74, 483-548 (1995) · Zbl 0853.35068 |
| [98] | 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 |
| [99] | Montecinos, G. I.; Müller, L. O.; Toro, E. F., Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes, J. Comput. Phys., 266, 101-123 (2014) · Zbl 1323.92065 |
| [100] | Montecinos, G. I.; Toro, E. F., Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes, J. Comput. Phys., 275, 415-442 (2014) · Zbl 1349.65378 |
| [101] | Müller, B., High order numerical simulation of aeolian tones, Comput. Fluids, 37, 450-462 (2008) · Zbl 1237.76106 |
| [102] | Müller, L. O.; Parés, C.; Toro, E. F., Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties, J. Comput. Phys., 242, 53-85 (2013) · Zbl 1323.92066 |
| [103] | Müller, L. O.; Toro, E. F., Well-balanced high-order solver for blood flow in networks of vessels with variable properties, Int. J. Numer. Methods Biomed. Eng., 29, 12, 1388-1411 (2013) |
| [104] | Muracchini, A.; Ruggeri, T.; Seccia, L., Dispersion relation in the high frequency limit and non linear wave stability for hyperbolic dissipative systems, Wave Motion, 15, 2, 143-158 (1992) · Zbl 0775.35004 |
| [105] | Ndanou, S.; Favrie, N.; Gavrilyuk, S., Criterion of hyperbolicity in hyperelasticity in the case of the stored energy in separable form, J. Elast., 115, 1, 1-25 (2014) · Zbl 1302.35254 |
| [106] | Nishikawa, H., A first-order system approach for diffusion equation. I: second-order residual-distribution schemes, J. Comput. Phys., 227, 1, 315-352 (2007) · Zbl 1127.65062 |
| [107] | Nishikawa, H., A first-order system approach for diffusion equation. II: unification of advection and diffusion, J. Comput. Phys., 229, 11, 3989-4016 (2010) · Zbl 1192.65116 |
| [108] | Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal., 44, 300-321 (2006) · Zbl 1130.65089 |
| [109] | Peshkov, I.; Grmela, M.; Romenski, E., Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid-fluid transitions, Contin. Mech. Thermodyn., 27, 6, 905-940 (November 2015) · Zbl 1341.80023 |
| [110] | Peshkov, I.; Romenski, E., A hyperbolic model for viscous Newtonian flows, Contin. Mech. Thermodyn., 28, 85-104 (2016) · Zbl 1348.76046 |
| [111] | Prandtl, L., Über Flüssigkeitsbewegung bei sehr kleiner Reibung, (Verhandlg. III. Intern. Math. Kongr.. Verhandlg. III. Intern. Math. Kongr., Heidelberg (1904)), 484-491 · JFM 36.0800.02 |
| [112] | Qiu, J.; Dumbser, M.; Shu, C. W., The discontinuous Galerkin method with Lax-Wendroff type time discretizations, Comput. Methods Appl. Mech. Eng., 194, 4528-4543 (2005) · Zbl 1093.76038 |
| [113] | Rhebergen, S.; Bokhove, O.; van der Vegt, J. J.W., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys., 227, 1887-1922 (2008) · Zbl 1153.65097 |
| [114] | Romenski, E.; Drikakis, D.; Toro, E., Conservative models and numerical methods for compressible two-phase flow, J. Sci. Comput., 42, 1, 68-95 (2010) · Zbl 1203.76095 |
| [115] | Romenski, E.; Resnyansky, A. D.; Toro, E. F., Conservative hyperbolic model for compressible two-phase flow with different phase pressures and temperatures, Q. Appl. Math., 65, 2, 259-279 (2007) · Zbl 1145.35430 |
| [116] | Romenski, E. I., Hyperbolic equations of maxwell’s nonlinear model of elastoplastic heat-conducting media, Sib. Math. J., 30, 4, 606-625 (1989) · Zbl 0741.73022 |
| [117] | Romenski, E. I., Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Math. Comput. Model., 28, 10, 115-130 (1998) · Zbl 1076.74501 |
| [118] | Romenski, E. I., Thermodynamics and hyperbolic systems of balance laws in continuum mechanics, (Toro, E. F., Godunov Methods (2001), Springer), 745-761 · Zbl 1017.74004 |
| [119] | Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR, 1, 267-279 (1961) |
| [120] | Xin, Z.; Jin, S., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48, 235-277 (1995) · Zbl 0826.65078 |
| [121] | Schlichting, H.; Gersten, K., Grenzschichttheorie (2005), Springer Verlag |
| [122] | Schwartzkopff, T.; Munz, C. D.; Toro, E. F., ADER: a high order approach for linear hyperbolic systems in 2d, J. Sci. Comput., 17, 1-4, 231-240 (2002) · Zbl 1022.76034 |
| [123] | Sette, D.; Busala, A.; Hubbard, J. C., Energy transfer by collisions in vapors of chlorinated methanes, J. Chem. Phys., 23, 787-793 (1955) |
| [124] | Shu, C. W.; Don, W. S.; Gottlieb, D.; Schilling, O.; Jameson, L., Numerical convergence study of nearly incompressible, inviscid Taylor-Green vortex flow, J. Sci. Comput., 24, 1-27 (2005) · Zbl 1161.76535 |
| [125] | 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 (2014), Springer), 945-953 · Zbl 1426.76429 |
| [126] | Stroud, A. H., Approximate Calculation of Multiple Integrals (1971), Prentice-Hall Inc.: Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013 |
| [127] | Taube, A.; Dumbser, M.; Balsara, D.; Munz, C. D., Arbitrary high order discontinuous Galerkin schemes for the magnetohydrodynamic equations, J. Sci. Comput., 30, 441-464 (2007) · Zbl 1176.76075 |
| [128] | Tavelli, M.; Dumbser, M., A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier-Stokes equations, Appl. Math. Comput., 248, 70-92 (2014) · Zbl 1338.76068 |
| [129] | Tavelli, M.; Dumbser, M., A staggered space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations on two-dimensional triangular meshes, Comput. Fluids, 119, 235-249 (2015) · Zbl 1390.76360 |
| [130] | Titarev, V. A.; Romenski, E. I.; Toro, E. F., MUSTA-type upwind fluxes for non-linear elasticity, Int. J. Numer. Methods Eng., 73, 897-926 (2008) · Zbl 1159.74046 |
| [131] | Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 1-4, 609-618 (December 2002) · Zbl 1024.76028 |
| [132] | Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736 (2005) · Zbl 1060.65641 |
| [133] | Titarev, V. A.; Tsoutsanis, P.; Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8, 585-609 (2010) · Zbl 1364.76121 |
| [134] | Toro, E. F.; Titarev, V. A., Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys., 212, 1, 150-165 (2006) · Zbl 1087.65590 |
| [135] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (2009), Springer · Zbl 1227.76006 |
| [136] | Toro, E. F.; Hidalgo, A., ADER finite volume schemes for nonlinear reaction-diffusion equations, Appl. Numer. Math., 59, 73-100 (2009) · Zbl 1155.65065 |
| [137] | Toro, E. F.; Millington, R. C.; Nejad, L. A.M., Towards very high order Godunov schemes, (Toro, E. F., Godunov Methods. Theory and Applications (2001), Kluwer/Plenum Academic Publishers), 905-938 · Zbl 0989.65094 |
| [138] | Toro, E. F.; Montecinos, G. I., Advection-diffusion-reaction equations: hyperbolization and high-order ADER discretizations, SIAM J. Sci. Comput., 36, 5, A2423-A2457 (2014) · Zbl 1307.65117 |
| [139] | Toro, E. F.; Montecinos, G. I., Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws, J. Comput. Phys., 303, 146-172 (2015) · Zbl 1349.76399 |
| [140] | Toro, E. F.; Titarev, V. A., Solution of the generalized Riemann problem for advection-reaction equations, (Proc. Roy. Soc. London (2002)), 271-281 · Zbl 1019.35061 |
| [141] | Tsoutsanis, P.; Titarev, V. A.; Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230, 1585-1601 (2011) · Zbl 1210.65160 |
| [142] | Wallace, Duane C., Statistical mechanics of monatomic liquids, Phys. Rev. E, 56, 4, 4179 (1997) |
| [143] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173 (1984) · Zbl 0573.76057 |
| [144] | Yan, J.; Shu, C. W., A local discontinuous Galerkin method for KdV-type equations, SIAM J. Numer. Anal., 40, 769-791 (2002) · Zbl 1021.65050 |
| [145] | Yan, J.; Shu, C. W., Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. Sci. Comput., 17, 1-4, 27-47 (2002) · Zbl 1003.65115 |
| [146] | Zanotti, O.; Dumbser, M., A high order special relativistic hydrodynamic and magnetohydrodynamic code with space-time adaptive mesh refinement, Comput. Phys. Commun., 188, 110-127 (2015) · Zbl 1344.76058 |
| [147] | Zanotti, O.; Dumbser, M., Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables, Comput. Astrophys. Cosmol., 3, 1-32 (2016) |
| [148] | Zanotti, O.; Fambri, F.; Dumbser, M., Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement, Mon. Not. R. Astron. Soc., 452, 3010-3029 (2015) |
| [149] | Zanotti, O.; Fambri, F.; Dumbser, M.; Hidalgo, A., Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori subcell finite volume limiting, Comput. Fluids, 118, 204-224 (2015) · Zbl 1390.76381 |
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.
