A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport

Manuel Jesús Castro Díaz; Enrique Domingo Fernández-Nieto; Tomás Morales de Luna; Gladys Narbona-Reina; Carlos Parés

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 47, Issue: 1, page 1-32
  • ISSN: 0764-583X

Abstract

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The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity preserving property of the corresponding numerical methods are presented. The numerical results provided by the two HLLC solvers are compared between them and also with those obtained with a Roe-type method in a number of 1d and 2d test problems. This comparison shows that, while the quality of the numerical solutions is comparable, the computational cost of the HLLC solvers is lower, as only some partial information of the eigenstructure of the matrix system is needed.

How to cite

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Castro Díaz, Manuel Jesús, et al. "A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport." ESAIM: Mathematical Modelling and Numerical Analysis 47.1 (2012): 1-32. <http://eudml.org/doc/222167>.

@article{CastroDíaz2012,
abstract = {The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity preserving property of the corresponding numerical methods are presented. The numerical results provided by the two HLLC solvers are compared between them and also with those obtained with a Roe-type method in a number of 1d and 2d test problems. This comparison shows that, while the quality of the numerical solutions is comparable, the computational cost of the HLLC solvers is lower, as only some partial information of the eigenstructure of the matrix system is needed.},
author = {Castro Díaz, Manuel Jesús, Fernández-Nieto, Enrique Domingo, Morales de Luna, Tomás, Narbona-Reina, Gladys, Parés, Carlos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Well-balanced; finite volume method; path-conservative; simple Riemann solver; HLLC; well-balanced},
language = {eng},
month = {7},
number = {1},
pages = {1-32},
publisher = {EDP Sciences},
title = {A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport},
url = {http://eudml.org/doc/222167},
volume = {47},
year = {2012},
}

TY - JOUR
AU - Castro Díaz, Manuel Jesús
AU - Fernández-Nieto, Enrique Domingo
AU - Morales de Luna, Tomás
AU - Narbona-Reina, Gladys
AU - Parés, Carlos
TI - A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/7//
PB - EDP Sciences
VL - 47
IS - 1
SP - 1
EP - 32
AB - The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity preserving property of the corresponding numerical methods are presented. The numerical results provided by the two HLLC solvers are compared between them and also with those obtained with a Roe-type method in a number of 1d and 2d test problems. This comparison shows that, while the quality of the numerical solutions is comparable, the computational cost of the HLLC solvers is lower, as only some partial information of the eigenstructure of the matrix system is needed.
LA - eng
KW - Well-balanced; finite volume method; path-conservative; simple Riemann solver; HLLC; well-balanced
UR - http://eudml.org/doc/222167
ER -

References

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  1. R. Abgrall and S. Karni, A comment on the computation of non-conservative products. J. Comput. Phys.229 (2010) 2759–2763.  
  2. M.S. Altinaker, W.H. Graf and E. Hopfinger, Flow structure in turbidity currents. J. Hydr. Res.34 (1996) 713–718.  
  3. F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, in Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004).  
  4. S.F. Bradford and N.D. Katopodes, Hydrodynamics of turbid underflows. i : Formulation and numerical analysis. J. Hydr. Eng.125 (1999) 1006–1015.  
  5. M.J. Castro, P.G. LeFloch, M.L. Muñoz-Ruiz and C. Parés, Why many theories of shock waves are necessary : Convergence error in formally path-consistent schemes. J. Comput. Phys.227 (2008) 8107–8129.  
  6. M.J. Castro, E.D. Fernández-Nieto, A.M. Ferreiro, J.A. García-Rodríguez and C. Parés, High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems. J. Sci. Comput.39 (2009) 67–114.  
  7. M. Castro Díaz, E. Fernéndez-Nieto and A. Ferreiro, Sediment transport models in shallow water equations and numerical approach by high order finite volume methods. Comput. Fluids37 (2008) 299–316.  
  8. M.J. Castro Díaz, E.D. Fernández-Nieto, A.M. Ferreiro and C. Parés, Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes. Comput. Methods Appl. Mech. Eng.198 (2009) 2520–2538.  
  9. S. Cordier, M. Le and T. Morales de Luna, Bedload transport in shallow water models : Why splitting (may) fail, how hyperbolicity (can) help. Adv. Water Resour.34 (2011) 980–989.  
  10. G. Dal Maso, P.G. Lefloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl.74 (1995) 483–548.  
  11. F. Exner, Über die wechselwirkung zwischen wasser und geschiebe in flüssen. Sitzungsber., Akad. WissenschaftenIIa (1925).  
  12. E.D. Fernández-Nieto, Modelling and numerical simulation of submarine sediment shallow flows : transport and avalanches. Bol. Soc. Esp. Mat. Apl. S􏿻 MA49 (2009) 83–103.  
  13. A.C. Fowler, N. Kopteva and C. Oakley, The formation of river channels. SIAM J. Appl. Math.67 (2007) 1016–1040.  
  14. J. Gallardo, S. Ortega, M. de la Asunción and J. Mantas, Two-Dimensional compact third-order polynomial reconstructions. solving nonconservative hyperbolic systems using GPUs. J. Sci. Comput.48 (2011) 141–163.  
  15. A. Grass, Sediment transport by waves and currents. SERC London Cent. Mar. Technol. Report No. FL29 (1981).  
  16. A. Harten, P.D. Lax and B. van Leer, On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Rev.25 (1983).  
  17. T.Y. Hou and P.G. Le Floch, Why nonconservative schemes converge to wrong solutions : error analysis. Math. Comput.62 (1994) 497–530.  
  18. S.M. Khan, J. Imran, S. Bradford and J. Syvitski, Numerical modeling of hyperpycnal plume. Mar. Geol.222-223 (2005) 193–211.  
  19. Y. Kubo, Experimental and numerical study of topographic effects on deposition from two-dimensional, particle-driven density currents. Sediment. Geol.164 (2004) 311–326.  
  20. Y. Kubo and T. Nakajima, Laboratory experiments and numerical simulation of sediment-wave formation by turbidity currents. Mar. Geol.192 (2002) 105–121.  
  21. D.A. Lyn and M. Altinakar, St. Venant-Exner equations for Near-Critical and transcritical flows. J. Hydr. Eng.128 (2002) 579–587.  
  22. E. Meyer-Peter, and R. Müller, Formulas for bed-load transport, in 2nd meeting IAHSR. Stockholm, Sweden (1948) 1–26.  
  23. T. Morales de Luna, M.J. Castro Díaz, C. Parés Madroñal and E.D. Fernández Nieto, On a shallow water model for the simulation of turbidity currents. Commun. Comput. Phys.6 (2009) 848–882.  
  24. T. Morales de Luna, M.J. Castro Díaz and C. Parés Madroñal, A duality method for sediment transport based on a modified Meyer-Peter & Müller model. J. Sci. Comput.48 (2010) 258–273.  
  25. P.H. Morris and D.J. Williams, Relative celerities of mobile bed flows with finite solids concentrations. J. Hydr. Eng.122 (1996) 311–315.  
  26. M.L. Muñoz Ruiz and C. Parés, On the convergence and Well-Balanced property of Path-conservative numerical schemes for systems of balance laws. J. Sci. Comput.48 (2011) 274–295.  
  27. P. Nielsen, Coastal Bottom Boundary Layers and Sediment Transport. World Scientific Pub. Co. Inc. (1992).  
  28. C. Parés, Numerical methods for nonconservative hyperbolic systems : a theoretical framework. SIAM J. Numer. Anal.44 (2006) 300–321 (electronic).  
  29. C. Parés and M.L. Muñoz Ruiz, On some difficulties of the numerical approximation of nonconservative hyperbolic systems. Bol. Soc. Esp. Mat. Apl.47 (2009) 23–52.  
  30. G. Parker, Y. Fukushima and H.M. Pantin, Self-accelerating turbidity currents. J. Fluid Mech.171 (1986) 145–181.  
  31. E.F. Toro, M. Spruce and W. Speares, Restoration of the contact surface in the HLL-Riemann solver. Shock Waves4 (1994) 25–34.  
  32. E.F. Toro, Shock-capturing methods for free-surface shallow flows. John Wiley (2001).  
  33. L. Van Rijn, Sediment transport : bed load transport. J. Hydr. Eng.110 (1984) 1431–1456. 

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