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An exactly curl-free finite-volume scheme for a hyperbolic compressible barotropic two-phase model (2403.18724v1)

Published 27 Mar 2024 in math.NA and cs.NA

Abstract: We present a new second order accurate structure-preserving finite volume scheme for the solution of the compressible barotropic two-phase model of Romenski et. al in multiple space dimensions. The governing equations fall into the wider class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems and consist of a set of first-order hyperbolic partial differential equations (PDE). In the absence of algebraic source terms, the model is subject to a curl-free constraint for the relative velocity between the two phases. The main objective of this paper is, therefore, to preserve this structural property exactly also at the discrete level. The new numerical method is based on a staggered grid arrangement where the relative velocity field is stored in the cell vertexes while all the remaining variables are stored in the cell centers. This allows the definition of discretely compatible gradient and curl operators, which ensure that the discrete curl errors of the relative velocity field remain zero up to machine precision. A set of numerical results confirms this property also experimentally.

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References (64)
  1. A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics. Applied Mathematics and Computation, 440:127629, 2023.
  2. Towards a gauge-polyvalent numerical relativity code. Physical Review D, 79(4):044026, 2009.
  3. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. International Journal of Multiphase Flow, 12(6):861–889, 1986.
  4. Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design. Communications in Applied Mathematics and Computational Science, 5(1):235–294, 2023.
  5. D. Balsara and D. Spicer. 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.
  6. D. S. Balsara. Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction. The Astrophysical Journal Supplement Series, 151(1):149, 2004.
  7. A high order semi-implicit scheme for ideal magnetohydrodynamics. In E. Franck, J. Fuhrmann, V. Michel-Dansac, and L. Navoret, editors, Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems, pages 21–37, Cham, 2023. Springer Nature Switzerland.
  8. Locally structure-preserving div-curl operators for high order discontinuous Galerkin schemes. Journal of Computational Physics, 486:112130, 2023.
  9. A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics. Journal of Computational Physics, 424, 2021.
  10. A simulation study of east-west IMF effects on the magnetosphere. Geophysical Research Letters, 8(4):397–400, 1981.
  11. Numerical simulations with a first-order BSSN formulation of Einstein’s field equations. Physical Review D, 85(8):084004, 2012.
  12. S. Busto and M. Dumbser. A new thermodynamically compatible finite volume scheme for magnetohydrodynamics. SIAM Journal on Numerical Analysis, 61(1):343–364, 2023.
  13. On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems. Journal of Scientific Computing, 87, 2021.
  14. On thermodynamically compatible finite volume schemes for continuum mechanics. SIAM Journal on Scientific Computing, 44(3):A1723–A1751, 2022.
  15. Well-balanced high order extensions of Godunov’s method for semilinear balance laws. SIAM Journal of Numerical Analysis, 46:1012–1039, 2008.
  16. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow–water systems. Mathematics of Computation, 75:1103–1134, 2006.
  17. S. Chiocchetti and M. Dumbser. An exactly curl-free staggered semi-implicit finite volume scheme for a first order hyperbolic model of viscous two-phase flows with surface tension. Journal of Scientific Computing, 94, 2023.
  18. High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension. Journal of Computational Physics, 426:109898, 2021.
  19. W. Dai and P. R. Woodward. A simple finite difference scheme for multidimensional magnetohydrodynamical equations. Journal of Computational Physics, 142(2):331–369, 1998.
  20. Hyperbolic divergence cleaning for the MHD equations. Journal of Computational Physics, 175(2):645–673, 2002.
  21. A new approach to divergence cleaning in magnetohydrodynamic simulations. In Hyperbolic Problems: Theory, Numerics, Applications, pages 509––518. Springer Berlin Heidelberg, 01 2003.
  22. Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations. J. Comput. Phys., 364:420–467, 2018.
  23. C. R. DeVore. Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics. Journal of Computational Physics, 92(1):142–160, 1991.
  24. F. Dhaouadi and M. Dumbser. A first order hyperbolic reformulation of the Navier-Stokes-Korteweg system based on the GPR model and an augmented Lagrangian approach. Journal of Computational Physics, 470, 2022.
  25. F. Dhaouadi and M. Dumbser. A structure-preserving finite volume scheme for a hyperbolic reformulation of the Navier–Stokes–Korteweg equations. Mathematics, 11, 2023.
  26. F. Dhaouadi and S. Gavrilyuk. An eulerian hyperbolic model for heat transfer derived via Hamilton’s principle: analytical and numerical study. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 480(2283):20230440, 2024.
  27. M. Dumbser. A simple two-phase method for the simulation of complex free surface flows. Computer Methods in Applied Mechanics and Engineering, 200(9):1204–1219, 2011.
  28. On Numerical Methods for Hyperbolic PDE with Curl Involutions, pages 125–134. Springer International Publishing, 2020.
  29. On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations. Journal of Computational Physics, 404:109088, 2020.
  30. FORCE schemes on unstructured meshes II: Non–conservative hyperbolic systems. Computer Methods in Applied Mechanics and Engineering, 199:625–647, 2010.
  31. Simulation of magnetohydrodynamic flows-a constrained transport method. The Astrophysical Journal, 332:659–677, 1988.
  32. An unsplit Godunov method for ideal MHD via constrained transport. Journal of Computational Physics, 205(2):509–539, 2005.
  33. S. K. Godunov. An interesting class of quasi-linear systems. Doklady Akademii Nauk SSSR, 139(3):521–523, 1961.
  34. S. K. Godunov. Symmetric form of the magnetohydrodynamic equation. Numerical Methods for Mechanics of Continuum Medium, 3(1):26–34, 1972.
  35. Nonstationary equations of the nonlinear theory of elasticity in Eulerian coordinates. Journal of Applied Mechanics and Technical Physics, 13:868–884, 1972.
  36. Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, 2003.
  37. Globally constraint-preserving FR/DG scheme for Maxwell’s equations at all orders. Journal of computational physics, 394:298–328, 2019.
  38. R. Holland. Finite-difference solution of Maxwell’s equations in generalized nonorthogonal coordinates. IEEE Transactions on Nuclear Science, 30(6):4589–4591, 1983.
  39. Implicit–explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning. Computer Physics Communications, 180(10):1760–1767, 2009.
  40. R. Jeltsch and M. Torrilhon. On curl–preserving finite volume discretizations for shallow water equations. BIT Numerical Mathematics, 46:S35–S53, 2006.
  41. An implicit-explicit solver for a two-fluid single-temperature model. Journal of Computational Physics, 498:112696, 2024.
  42. An all Mach number finite volume method for isentropic two–phase flow. Journal of Numerical Mathematics, 31(3):175–204, 2023.
  43. Divergence correction techniques for Maxwell solvers based on a hyperbolic model. Journal of Computational Physics, 161(2):484–511, 2000.
  44. C. Parés. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44:300–321, 2006.
  45. Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme. Computers & Fluids, 224:104963, 2021.
  46. Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mechanics and Thermodynamics, 30(6):1343–1378, 2018.
  47. I. Peshkov and E. I. Romenski. A hyperbolic model for viscous newtonian flows. Continuum Mechanics and Thermodynamics, 28:85–104, 2016.
  48. K. G. Powell. An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA, 1994.
  49. K. G. Powell. An Approximate Riemann Solver for Magnetohydrodynamics, pages 570–583. Springer Berlin Heidelberg, 1997.
  50. A solution-adaptive upwind scheme for ideal magnetohydrodynamics. Journal of Computational Physics, 154(2):284–309, 1999.
  51. L. Río-Martín and M. Dumbser. High-order ADER discontinuous Galerkin schemes for a symmetric hyperbolic model of compressible barotropic two-fluid flows. Communications on Applied Mathematics and Computation, 2023.
  52. E. I. Romenski. Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Mathematical and computer modelling, 28(10):115–130, 1998.
  53. Conservative models and numerical methods for compressible two-phase flow. Journal of Scientific Computing, 42:68–95, 2010.
  54. Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures. Quarterly of Applied Mathematics, 65(2):259–279, 2007.
  55. Compressible two-phase flows: Two-pressure models and numerical methods. Computational Fluid Dynamics Journal, 13, 2012.
  56. E. I. Romensky. Thermodynamics and Hyperbolic Systems of Balance Laws in Continuum Mechanics, pages 745–761. Springer US, New York, NY, 2001.
  57. V. Rusanov. The calculation of the interaction of non-stationary shock waves and obstacles. USSR Computational Mathematics and Mathematical Physics, 1(2):304–320, 1962.
  58. Exact and numerical solutions of the Riemann problem for a conservative model of compressible two–phase flows. Journal of Scientific Computing, 93(83), 2022.
  59. A. Thomann and M. Dumbser. Thermodynamically compatible discretization of a compressible two-fluid model with two entropy inequalities. Journal of Scientific Computing, 97(1):9, 2023.
  60. E. F. Toro. Riemann solvers and numerical methods for fluid dynamics. Springer, 2009.
  61. M. Torrilhon and M. Fey. Constraint-preserving upwind methods for multidimensional advection equations. SIAM Journal on Numerical Analysis, 42:1694–1728, 2004.
  62. G. Tóth. The ∇∇\nabla∇· B= 0 constraint in shock-capturing magnetohydrodynamics codes. Journal of Computational Physics, 161(2):605–652, 2000.
  63. Divergence-free WENO reconstruction-based finite volume scheme for solving ideal MHD equations on triangular meshes. Communications in Computational Physics, 19(4):841–880, 2016.
  64. K. Yee. Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media. IEEE Transactions on antennas and propagation, 14(3):302–307, 1966.
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