Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables

Springer Science and Business Media LLC - Tập 3 Số 1 - 2016
Olindo Zanotti1, Michael Dumbser1
1Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, Trento, 38123, Italy

Tóm tắt

AbstractWe present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR).

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Tài liệu tham khảo

Andrianov, N, Warnecke, G: The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 212, 434-464 (2004)

Anile, AM: Relativistic Fluids and Magneto-fluids. Cambridge University Press, Cambridge (1990)

Antón, L, Miralles, JA, Martí, JM, Ibáñez, JM, Aloy, MA, Mimica, P: Relativistic magnetohydrodynamics: renormalized eigenvectors and full wave decomposition Riemann solver. Astron. Astrophys. Suppl. Ser. 188, 1-31 (2010)

Baer, MR, Nunziato, JW: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flow 12, 861-889 (1986)

Balsara, D: Total variation diminishing scheme for relativistic magnetohydrodynamics. Astrophys. J. Suppl. Ser. 132, 83-101 (2001)

Balsara, D, Spicer, D: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149, 270-292 (1999)

Balsara, DS, 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)

Beckwith, K, Stone, JM: A second-order Godunov method for multi-dimensional relativistic magnetohydrodynamics. Astrophys. J. Suppl. Ser. 193(1), 6 (2011)

Bodo, G, Mignone, A, Rosner, R: Kelvin-Helmholtz instability for relativistic fluids. Phys. Rev. E 70(3), 036304 (2004)

Boscheri, W, Dumbser, M: Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Commun. Comput. Phys. 14, 1174-1206 (2013)

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)

Boscheri, W, Balsara, DS, Dumbser, M: Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers. J. Comput. Phys. 267, 112-138 (2014a)

Boscheri, W, Dumbser, M, Balsara, DS: High order Lagrangian ADER-WENO schemes on unstructured meshes - application of several node solvers to hydrodynamics and magnetohydrodynamics. Int. J. Numer. Methods Fluids 76, 737-778 (2014b)

Bucciantini, N, Del Zanna, L: General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: the X-ECHO code. Astron. Astrophys. 528, A101 (2011)

Buchmüller, P, Helzel, C: Improved accuracy of high-order WENO finite volume methods on Cartesian grids. J. Sci. Comput. 61(2), 343-368 (2014)

Buchmüller, P, Dreher, J, Helzel, C: Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement. Appl. Math. Comput. (2015). doi:10.1016/j.amc.2015.03.078

Castro, MJ, Gallardo, JM, 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)

Castro, MJ, Gallardo, JM, López, JA, Parés, C: Well-balanced high order extensions of Godunov’s method for semilinear balance laws. SIAM J. Numer. Anal. 46, 1012-1039 (2008a)

Castro, MJ, LeFloch, PG, Muñoz-Ruiz, ML, Parés, C: Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. J. Comput. Phys. 227(17), 8107-8129 (2008b)

Castro, MJ, Gallardo, JM, Marquina, A: Approximate Osher-Solomon schemes for hyperbolic systems. Appl. Math. Comput. (2015). doi:10.1016/j.amc.2015.06.104

Clain, S, Diot, S, Loubère, R: A high-order finite volume method for systems of conservation laws - multi-dimensional optimal order detection (MOOD). J. Comput. Phys. 230(10), 4028-4050 (2011)

Dal Maso, G, LeFloch, PG, Murat, F: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483-548 (1995)

Dedner, A, Kemm, F, Kröner, D, Munz, CD, Schnitzer, T, Wesenberg, M: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175, 645-673 (2002)

Del Zanna, L, Bucciantini, N, Londrillo, P: An efficient shock-capturing central-type scheme for multidimensional relativistic flows. II. Magnetohydrodynamics. Astron. Astrophys. 400, 397-413 (2003)

Del Zanna, L, Zanotti, O, Bucciantini, N, Londrillo, P: ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics. Astron. Astrophys. 473, 11-30 (2007)

Deledicque, V, Papalexandris, MV: An exact Riemann solver for compressible two-phase flow models containing non-conservative products. J. Comput. Phys. 222, 217-245 (2007)

Diot, S, Clain, S, Loubère, R: Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids 64, 43-63 (2012)

Dumbser, M: Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput. Fluids 39, 60-76 (2010)

Dumbser, M: A diffuse interface method for complex three-dimensional free surface flows. Comput. Methods Appl. Mech. Eng. 257, 47-64 (2013)

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)

Dumbser, M, Munz, C: Building blocks for arbitrary high order discontinuous Galerkin schemes. J. Sci. Comput. 27, 215-230 (2006)

Dumbser, M, Toro, EF: A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput. 48, 70-88 (2011a)

Dumbser, M, Toro, EF: On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Commun. Comput. Phys. 10, 635-671 (2011b)

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)

Dumbser, M, Balsara, DS, Toro, EF, Munz, C-D: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227, 8209-8253 (2008a)

Dumbser, M, Enaux, C, Toro, EF: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971-4001 (2008b)

Dumbser, M, Castro, M, Parés, C, Toro, EF: ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows. Comput. Fluids 38, 1731-1748 (2009)

Dumbser, M, Hidalgo, A, Castro, M, Parés, C, Toro, EF: FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng. 199, 625-647 (2010)

Dumbser, M, Uuriintsetseg, A, Zanotti, O: On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws. Commun. Comput. Phys. 14, 301-327 (2013)

Dumbser, M, Zanotti, O, Hidalgo, A, Balsara, DS: ADER-WENO finite volume schemes with space-time adaptive mesh refinement. J. Comput. Phys. 248, 257-286 (2013)

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)

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)

Giacomazzo, B, Rezzolla, L: The exact solution of the Riemann problem in relativistic MHD. J. Fluid Mech. 562, 223-259 (2006)

Hidalgo, A, Dumbser, M: ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. J. Sci. Comput. 48, 173-189 (2011)

Hu, C, Shu, C: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97-127 (1999)

Jiang, G, Shu, C: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228 (1996)

Jin, S, Xin, Z: Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, and relaxation schemes. SIAM J. Numer. Anal. 35(6), 2385-2404 (1998)

Kim, J, Balsara, DS: A stable HLLC Riemann solver for relativistic magnetohydrodynamics. J. Comput. Phys. 270, 634-639 (2014)

Komissarov, SS: On the properties of Alfvén waves in relativistic magnetohydrodynamics. Phys. Lett. A 232, 435-442 (1997)

Komissarov, SS: A Godunov-type scheme for relativistic magnetohydrodynamics. Mon. Not. R. Astron. Soc. 303, 343-366 (1999)

Leismann, T, Antón, L, Aloy, MA, Müller, E, Martí, JM, Miralles, JA, Ibáñez, JM: Relativistic MHD simulations of extragalactic jets. Astron. Astrophys. 436, 503-526 (2005)

Londrillo, P, Del Zanna, L: High-order upwind schemes for multidimensional magnetohydrodynamics. Astrophys. J. 530, 508-524 (2000)

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)

Mignone, A, Bodo, G: An HLLC Riemann solver for relativistic flows - I. Hydrodynamics. Mon. Not. R. Astron. Soc. 364, 126-136 (2005)

Mignone, A, Ugliano, M, Bodo, G: A five-wave Harten-Lax-van Leer Riemann solver for relativistic magnetohydrodynamics. Mon. Not. R. Astron. Soc. 393(4), 1141-1156 (2009)

Müller, LO, Parés, C, Toro, EF: Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties. J. Comput. Phys. 242, 53-85 (2013)

Muñoz, ML, Parés, C: Godunov method for nonconservative hyperbolic systems. Math. Model. Numer. Anal. 41, 169-185 (2007)

Munz, CD: On the construction and comparison of two-step schemes for the Euler equations. Notes Numer. Fluid Mech. 14, 195-217 (1986)

Murrone, A, Guillard, H: A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202, 664-698 (2005)

Noble, SC, Gammie, CF, McKinney, JC, Del Zanna, L: Primitive variable solvers for conservative general relativistic magnetohydrodynamics. Astrophys. J. 641, 626-637 (2006)

Noh, W: Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys. 72(1), 78-120 (1987)

Palenzuela, C, Lehner, L, Reula, O, Rezzolla, L: Beyond ideal MHD: towards a more realistic modelling of relativistic astrophysical plasmas. Mon. Not. R. Astron. Soc. 394, 1727-1740 (2009)

Pares, C: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44(1), 300-321 (2006)

Parés, C, Castro, MJ: On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. Math. Model. Numer. Anal. 38, 821-852 (2004)

Pareschi, L, Puppo, G, Russo, G: Central Runge-Kutta schemes for conservation laws. SIAM J. Sci. Comput. 26(3), 979-999 (2005)

Perucho, M, Lobanov, AP, Martí, J-M, Hardee, PE: The role of Kelvin-Helmholtz instability in the internal structure of relativistic outflows. The case of the jet in 3C 273. Astron. Astrophys. 456, 493-504 (2006)

Perucho, M, Hanasz, M, Martí, J-M, Miralles, J-A: Resonant Kelvin-Helmholtz modes in sheared relativistic flows. Phys. Rev. E 75(5), 056312 (2007)

Pidatella, R, Puppo, G, Russo, G, Santagati, P: Semi-conservative schemes for conservation laws. J. Sci. Comput. (2016, to appear)

Radice, D, Rezzolla, L: Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes. Phys. Rev. D 84(2), 024010 (2011)

Radice, D, Rezzolla, L: THC: a new high-order finite-difference high-resolution shock-capturing code for special-relativistic hydrodynamics. Astron. Astrophys. 547, A26 (2012)

Rezzolla, L, Zanotti, O: An improved exact Riemann solver for relativistic hydrodynamics. J. Fluid Mech. 449, 395-411 (2001)

Rezzolla, L, Zanotti, O: Relativistic Hydrodynamics. Oxford University Press, Oxford (2013)

Saurel, R, Abgrall, R: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425-467 (1999)

Schwendeman, DW, Wahle, CW, Kapila, AK: The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212, 490-526 (2006)

Sedov, LI: Similarity and Dimensional Methods in Mechanics (1959)

Sod, GA: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1-31 (1978)

Stroud, A: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)

Taube, A, Dumbser, M, Balsara, D, Munz, C: Arbitrary high order discontinuous Galerkin schemes for the magnetohydrodynamic equations. J. Sci. Comput. 30, 441-464 (2007)

Teukolsky, SA: Formulation of discontinuous Galerkin methods for relativistic astrophysics. arXiv:1510.01190

Titarev, V, Toro, E: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17(1-4), 609-618 (2002)

Titarev, VA, Toro, EF: ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys. 204, 715-736 (2005)

Toro, E, Titarev, V: Solution of the generalized Riemann problem for advection-reaction equations. Proc. R. Soc. Lond. 458, 271-281 (2002)

Toro, E, Millington, R, Nejad, L: Towards very high order Godunov schemes. In: Toro, E (ed.) Godunov Methods. Theory and Applications, pp. 905-938. Kluwer/Plenum Academic Publishers, New York (2001)

Toro, EF: Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd edn. Springer, Berlin (1999)

Toro, EF, Hidalgo, A: ADER finite volume schemes for nonlinear reaction-diffusion equations. Appl. Numer. Math. 59, 73-100 (2009)

Toro, EF, Titarev, VA: Derivative Riemann solvers for systems of conservation laws and ADER methods. J. Comput. Phys. 212(1), 150-165 (2006)

Woodward, P, Colella, P: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115-173 (1984)

Zanotti, O, Dumbser, M: Numerical simulations of high Lundquist number relativistic magnetic reconnection. Mon. Not. R. Astron. Soc. 418, 1004-1011 (2011)

Zanotti, O, Dumbser, M: High order numerical simulations of the Richtmyer- Meshkov instability in a relativistic fluid. Phys. Fluids 27(7), 074105 (2015)

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)

Zanotti, O, Roedig, C, Rezzolla, L, Del Zanna, L: General relativistic radiation hydrodynamics of accretion flows - I. Bondi-Hoyle accretion. Mon. Not. R. Astron. Soc. 417, 2899-2915 (2011)

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 (2015a)

Zanotti, O, Fambri, F, Dumbser, M, Hidalgo, A: Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Comput. Fluids 118, 204-224 (2015b)

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