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A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. (English) Zbl 1147.65075
Summary: In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree \(N\) are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree \(M\geqslant N\), which are reconstructed from the underlying polynomials of degree \(N\). This approach yields a general, unified framework that contains as two special cases classical high order finite volume schemes \((N=0)\) as well as the usual discontinuous Galerkin (DG) method \((N=M)\). In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing \(N\neq 0\) and \(M>N\). The reconstruction operator is implemented for arbitrary polynomial degrees \(N\) and \(M\) on unstructured triangular and tetrahedral meshes in two and three space dimensions.
To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree \(M\) are evolved in time locally inside each element using a new local continuous space-time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space-time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy.
Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes \((N\neq 0,M>N)\) is computationally more efficient than classical finite volume or DG schemes. Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order weighted essentially non-oscillatory (WENO) finite volume discretization in space to cope with discontinuous solutions.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics, general
76W05 Magnetohydrodynamics and electrohydrodynamics
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
74S10 Finite volume methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
Software:
ECHO
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[1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, Journal of computational physics, 144, 45-58, (1994) · Zbl 0822.65062
[2] Atkins, H.; Shu, C.W., Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations, AIAA journal, 36, 775-782, (1998)
[3] Balsara, D., Total variation diminishing scheme for relativistic magneto-hydrodynamics, The astrophysical journal supplement series, 132, 83-101, (2001)
[4] Balsara, D., Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, The astrophysical journal supplement series, 151, 149-184, (2004)
[5] Balsara, D.; Altmann, C.; Munz, C.D.; Dumbser, M., A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes, Journal of computational physics, 226, 586-620, (2007) · Zbl 1124.65072
[6] Balsara, D.; Spicer, D., Maintaining pressure positivity in magneto-hydrodynamic simulations, Journal of computational physics, 148, 133-148, (1999) · Zbl 0930.76050
[7] Balsara, D.; Spicerm, D., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, Journal of computational physics, 149, 270-292, (1999) · Zbl 0936.76051
[8] T.J. Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper No. 90-0013, 28th Aerospace Sciences Meeting, January 1990.
[9] Ben-Artzi, M.; Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, Journal of computational physics, 55, 1-32, (1984) · Zbl 0535.76070
[10] Bourgeade, A.; LeFloch, P.; Raviart, P.A., An asymptotic expansion for the solution of the generalized Riemann problem. part II: application to the gas dynamics equations, Annales de l’institut Henri Poincaré (C) analyse non linéaire, 6, 437-480, (1989) · Zbl 0703.35106
[11] Brio, M.; Wu, C.C., An upwind differencing scheme for the equations of ideal magnetohydrodynamics, Journal of computational physics, 75, 400-422, (1988) · Zbl 0637.76125
[12] Castro, C.C.; Toro, E.F., Solvers for the high-order Riemann problem for hyperbolic balance laws, Journal of computational physics, 227, 2481-2513, (2008) · Zbl 1148.65066
[13] Cockburn, B.; Hou, S.; Shu, C.W., The runge – kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Mathematics of computation, 54, 545-581, (1990) · Zbl 0695.65066
[14] Cockburn, B.; Karniadakis, G.E.; Shu, C.W., Discontinuous Galerkin methods, Lecture notes in computational science and engineering, (2000), Springer
[15] 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, Journal of computational physics, 84, 90-113, (1989) · Zbl 0677.65093
[16] Cockburn, B.; Luskin, M.; Shu, C.W.; Suli, E., Enhanced accuracy by post-processing for finite element methods for hyperbolic equations, Mathematics of computation, 72, 577-606, (2003) · Zbl 1015.65049
[17] Cockburn, B.; Shu, C.W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of computation, 52, 411-435, (1989) · Zbl 0662.65083
[18] Cockburn, B.; Shu, C.W., The runge – kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Mathematical modelling and numerical analysis, 25, 337-361, (1991) · Zbl 0732.65094
[19] Cockburn, B.; Shu, C.W., The local discontinuous Galerkin method for time-dependent convection diffusion systems, SIAM journal on numerical analysis, 35, 2440-2463, (1998) · Zbl 0927.65118
[20] Cockburn, B.; Shu, C.W., The runge – kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of computational physics, 141, 199-224, (1998) · Zbl 0920.65059
[21] Cockburn, B.; Shu, C.W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, Journal of scientific computing, 16, 173-261, (2001) · Zbl 1065.76135
[22] Dahlburg, R.B.; Picone, J.M., Evolution of the orszag – tang vortex system in a compressible medium. I. initial average subsonic flow, Phys. fluids B, 1, 2153-2171, (1989)
[23] Dedner, A.; Kemm, F.; Kröner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, Journal of computational physics, 175, 645-673, (2002) · Zbl 1059.76040
[24] Dubiner, M., Spectral methods on triangles and other domains, Journal of scientific computing, 6, 345-390, (1991) · Zbl 0742.76059
[25] Dumbser, M., Arbitrary high order schemes for the solution of hyperbolic conservation laws in complex domains, (2005), Shaker Verlag Aachen
[26] Dumbser, M.; Enaux, C.; Toro, E.F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, Journal of computational physics, 227, 3971-4001, (2008) · Zbl 1142.65070
[27] Dumbser, M.; Käser, M., Arbitrary high order finite volume schemes for seismic wave propagation on unstructured meshes in 2d and 3d, Geophysical journal international, 171, 665-694, (2007)
[28] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, Journal of computational physics, 221, 693-723, (2007) · Zbl 1110.65077
[29] Dumbser, M.; Käser, M.; Titarev, V.A.; Toro, E.F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, Journal of computational physics, 226, 204-243, (2007) · Zbl 1124.65074
[30] Dumbser, M.; Käser, M.; Toro, E.F., An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes V: local time stepping and p-adaptivity, Geophysical journal international, 171, 695-717, (2007)
[31] Dumbser, M.; Munz, C.D., Arbitrary high order discontinuous Galerkin schemes, (), 295-333 · Zbl 1210.65165
[32] Dumbser, M.; Munz, C.D., Building blocks for arbitrary high order discontinuous Galerkin schemes, Journal of scientific computing, 27, 215-230, (2006) · Zbl 1115.65100
[33] Dumbser, M.; Schwartzkopff, T.; Munz, C.D., Arbitrary high order finite volume schemes for linear wave propagation, Notes on numerical fluid mechanics and multidisciplinary design (NNFM), (2006), Springer, pp. 129-144
[34] R.W. Dyson, Technique for very high order nonlinear simulation and validation, Technical Report TM-2001-210985, NASA, 2001.
[35] Le Floch, P.; Raviart, P.A., An asymptotic expansion for the solution of the generalized Riemann problem. part I: general theory, Annales de l’institut Henri Poincaré (C) analyse non linéaire, 5, 179-207, (1988) · Zbl 0679.35064
[36] Le Floch, P.; Tatsien, L., A global asymptotic expansion for the solution of the generalized Riemann problem, Annales de l’institut Henri Poincaré (C) analyse non linéaire, 3, 321-340, (1991) · Zbl 0731.35006
[37] Gassner, G.; Loercher, F.; Munz, C.D., A discontinuous Galerkin scheme based on a space – time expansion II. viscous flow equations in multi dimensions, Journal of scientific computing, 34, 260-286, (2008) · Zbl 1218.76027
[38] Giacomazzo, B.; Rezzolla, L., The exact solution of the Riemann problem in relativistic magnetohydrodynamics, Journal of fluid mechanics, 562, 223-259, (2006) · Zbl 1097.76073
[39] Godunov, S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Matematicheskij sbornik, 47, 271-306, (1959) · Zbl 0171.46204
[40] Godunov, S.K.; Romenski, E.I., Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates, Journal of applied mechanics and technical physics, 13, 868-885, (1972)
[41] Godunov, S.K.; Romenski, E.I., Thermodynamics, conservation laws, and symmetric forms of differential equations in mechanics of continuous media, (), 19-31 · Zbl 0875.73025
[42] Godunov, S.K.; Romenski, E.I., Elements of continuum mechanics and conservation laws, (2003), Kluwer Academic/Plenum Publishers
[43] Gottlieb, S.; Shu, C.W., Total variation diminishing runge – kutta schemes, Mathematics of computation, 67, 73-85, (1988) · Zbl 0897.65058
[44] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, Journal of computational physics, 71, 231-303, (1987) · Zbl 0652.65067
[45] Hirsch, C., Numerical computation of internal and external flows vol I: fundamentals of numerical discretisation, (1988), Wiley
[46] Honkkila, V.; Janhunen, P., HLLC solver for ideal relativistic MHD, Journal of computational physics, 223, 643-656, (2007) · Zbl 1111.76036
[47] Hu, C.; Shu, C.W., Weighted essentially non-oscillatory schemes on triangular meshes, Journal of computational physics, 150, 97-127, (1999) · Zbl 0926.65090
[48] Jiang, G.S.; Wu, C.C., A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics, Journal of computational physics, 150, 561-594, (1999) · Zbl 0937.76051
[49] Käser, M.; Iske, A., ADER schemes on adaptive triangular meshes for scalar conservation laws, Journal of computational physics, 205, 486-508, (2005) · Zbl 1072.65116
[50] Kolgan, V.P., Application of the minimum-derivative principle in the construction of finite-difference schemes for numerical analysis of discontinuous solutions in gas dynamics, Transactions of the central aerohydrodynamics institute, 3, 6, 68-77, (1972), (in Russian)
[51] Lax, P.D.; Wendroff, B., Systems of conservation laws, Communications in pure and applied mathematics, 13, 217-237, (1960) · Zbl 0152.44802
[52] Lindelöf, M.E., Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre, Comptes rendus hebdomadaires des séances de l’académie des sciences, 114, 454-457, (1894) · JFM 25.0509.01
[53] Loercher, F.; Gassner, G.; Munz, C.D., A discontinuous Galerkin scheme based on a space – time expansion. I. inviscid compressible flow in one space dimension, Journal of scientific computing, 32, 175-199, (2007) · Zbl 1143.76047
[54] Ollivier-Gooch, C.; Van Altena, M., A high-order-accurate unstructured mesh finite-volume scheme for the advection – diffusion equation, Journal of computational physics, 181, 729-752, (2002) · Zbl 1178.76251
[55] Orszag, S.A.; Tang, C.M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, Journal of fluid mechanics, 90, 129, (1979)
[56] Picone, J.M.; Dahlburg, R.B., Evolution of the orszag-Tang vortex system in a compressible medium. II. supersonic flow, Physics of fluids B, 3, 29-44, (1991)
[57] Qiu, J.; Dumbser, M.; Shu, C.W., The discontinuous Galerkin method with lax – wendroff type time discretizations, Computer methods in applied mechanics and engineering, 194, 4528-4543, (2005) · Zbl 1093.76038
[58] Qiu, J.; Shu, C.W., Finite difference WENO schemes with lax – wendroff type time discretization, SIAM journal on scientific computing, 24, 6, 2185-2198, (2003) · Zbl 1034.65073
[59] Qiu, J.; Shu, C.W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin method: one-dimensional case, Journal of computational physics, 193, 115-135, (2003) · Zbl 1039.65068
[60] Qiu, J.; Shu, C.W., Hermite WENO schemes and their application as limiters for runge – kutta discontinuous Galerkin method II: two dimensional case, Computers and fluids, 34, 642-663, (2005) · Zbl 1134.65358
[61] Rezzolla, L.; Zanotti, O., An improved exact Riemann solver for relativistic hydrodynamics, Journal of fluid mechanics, 449, 395-411, (2001) · Zbl 1009.76101
[62] Ryan, J.K.; Shu, C.W.; Atkins, H.L., Extension of a post-processing technique for the discontinuous Galerkin method for hyperbolic equations with applications to an aeroacoustic problem, SIAM journal on scientific computing, 26, 821-843, (2005) · Zbl 1137.65414
[63] Ryu, D.; Jones, T.W., Numerical magnetohydrodynamics in astrophysics: algorithm and tests for one-dimensional flow, Astrophysical journal, 442, 228-258, (1995)
[64] Schwartzkopff, T.; Dumbser, M.; Munz, C.D., Fast high order ADER schemes for linear hyperbolic equations, Journal of computational physics, 197, 532-539, (2004) · Zbl 1052.65078
[65] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, Journal of computational physics, 77, 439-471, (1988) · Zbl 0653.65072
[66] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, Journal of computational physics, 83, 32-78, (1989) · Zbl 0674.65061
[67] Stroud, A.H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013
[68] Taube, A.; Dumbser, M.; Balsara, D.; Munz, C.D., Arbitrary high order discontinuous Galerkin schemes for the magnetohydrodynamic equations, Journal of scientific computing, 30, 441-464, (2007) · Zbl 1176.76075
[69] Titarev, V.A.; Romenski, E.I.; Toro, E.F., MUSTA-type upwind fluxes for non-linear elasticity, International journal for numerical methods in engineering, 73, 897-926, (2008) · Zbl 1159.74046
[70] Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, Journal of scientific computing, 17, 1-4, 609-618, (2002) · Zbl 1024.76028
[71] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional nonlinear hyperbolic systems, Journal of computational physics, 204, 715-736, (2005) · Zbl 1060.65641
[72] Toro, E.F.; Titarev, V.A., Derivative Riemann solvers for systems of conservation laws and ADER methods, Journal of computational physics, 212, 1, 150-165, (2006) · Zbl 1087.65590
[73] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high order Godunov schemes, (), 905-938 · Zbl 0989.65094
[74] Toro, E.F.; Titarev, V.A., Solution of the generalized Riemann problem for advection – reaction equations, Proceedings of royal society of London, 271-281, (2002) · Zbl 1019.35061
[75] van der Vegt, J.J.W.; van der Ven, H., Space – time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. general formulation, Journal of computational physics, 182, 546-585, (2002) · Zbl 1057.76553
[76] van der Ven, H.; van der Vegt, J.J.W., Space – time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. efficient flux quadrature, Computer methods in applied mechanics and engineering, 191, 4747-4780, (2002) · Zbl 1099.76521
[77] van Leer, B., Towards the ultimate conservative difference scheme II: monotonicity and conservation combined in a second order scheme, Journal of computational physics, 14, 361-370, (1974) · Zbl 0276.65055
[78] van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to godunov’s method, Journal of computational physics, 32, 101-136, (1979) · Zbl 1364.65223
[79] van Leer, B., On the relationship between the upwind-differencing schemes of Godunov, engquist-osher and roe, SIAM journal on scientific and statistical computing, 5, 1-20, (1985) · Zbl 0547.65065
[80] B. van Leer, S. Nomura, Discontinuous Galerkin for diffusion, in: Proceedings of 17th AIAA Computational Fluid Dynamics Conference (June 6-9 2005), AIAA-2005-5108.
[81] Youssef, I.K.; El-Arabawy, H.A., Picard iteration algorithm combined with gauss – seidel technique for initial value problems, Applied mathematics and computation, 190, 345-355, (2007) · Zbl 1122.65369
[82] Del Zanna, L.; Bucciantini, N.; Londrillo, P., An efficient shock-capturing central-type scheme for multidimensional relativistic flows II. magnetohydrodynamics, Astronomy and astrophysics, 400, 397-413, (2003) · Zbl 1222.76122
[83] Del Zanna, L.; Zanotti, O.; Bucciantini, N.; Londrillo, P., ECHO: an Eulerian conservative high order scheme for general relativistic magnetohydrodynamics and magnetodynamics, Astronomy and astrophysics, 473, 11-30, (2007)
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