Numerical treatment of the loss of hyperbolicity of the two-layer shallow-water system

Journal of Scientific Computing

Volumn 48, Issue 1-3, 2011, Pages 16-40

  Castro Diaz M J ^a   Fernandez Nieto E D ^b   Gonzalez Vida J M ^a   Pares Madronal C ^a  

^a UNIVERSITY OF MÁLAGA   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

Complex eigenvalues; Finite volume method; Path conservative; Two layer shallow water

Indexed keywords

COMPLEX EIGENVALUES; FINITE VOLUME; FRICTION TERM; HYPERBOLICITY; INTENSE MIXING; LINEAR FRICTION; MECHANICAL ENERGIES; NUMERICAL SCHEME; NUMERICAL SOLUTION; NUMERICAL TESTS; NUMERICAL TREATMENTS; PATH-CONSERVATIVE; REAL FLOW; SEMI-IMPLICIT; SHALLOW WATERS; SHALLOW-WATER; SHEAR INSTABILITY; TWO LAYERS;

EIGENVALUES AND EIGENFUNCTIONS; FINITE VOLUME METHOD; FRICTION; TERNARY SYSTEMS; TRIBOLOGY; WATERWORKS;

SHEAR FLOW;

EID: 79959496523     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-010-9427-5     Document Type: Conference Paper

References (30)

1. R. Abgrall S. Karni 2009 Two-layer shallow water system: a relaxation approach SIAM J. Sci. Comput. 31 1603 1627 1188.76229 10.1137/06067167X
2. E. Audusse 2005 A multilayer Saint-Venant model: derivation and numerical validation Discrete Contin. Dyn. Syst.; Ser. B 5 189 214 1075.35030 10.3934/dcdsb.2005.5.189 (Pubitemid 40622760)
3. E. Audusse M.O. Bristeau 2007 Finite-volume solvers for a multilayer Saint-Venant system Int. J. Appl. Math. Comput. Sci. 17 311 319 1152.35305 10.2478/v10006-007-0025-0 (Pubitemid 47585305)
4. F. Bouchut T. Morales de Luna 2008 An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment Math. Model. Numer. Anal. 42 683 698 1203.76110 10.1051/m2an:2008019
5. F. Bouchut V. Zeitlin 2010 A robust well-balanced scheme for multi-layer shallow water equations Discrete Contin. Dyn Syst.; Ser. B 13 4 739 758 05715572 10.3934/dcdsb.2010.13.739
6. M.J. Castro J. Macías C. Parés 2001 A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system Math. Model. Numer. Anal. 35 1 107 127 1094.76046 10.1051/m2an:2001108
7. M.J. Castro J.A. García J.M. González J. Macías C. Parés M.E. Vázquez 2004 Numerical simulation of two layer shallow water flows through channels with irregular geometry J. Comput. Phys. 195 1 202 235 1087.76077 10.1016/j.jcp.2003.08.035 (Pubitemid 38354645)
8. M.J. Castro A.M. Ferreiro J.A. García J.M. González J. Macías C. Parés M.E. Vázquez 2005 On the numerical treatment of wet/dry fronts in shallow flows: applications to one-layer and two-layer systems Math. Comput. Model. 42 3-4 419 439 1121.76008 10.1016/j.mcm.2004.01.016 (Pubitemid 41454935)
9. M.J. Castro J.J. Gallardo C. Parés 2006 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 1096.65082 10.1090/S0025-5718-06-01851-5 (Pubitemid 44116289)
10. M.J. Castro J.A. García J.M. González C. Parés 2006 A parallel 2d finite volume scheme for solving systems of balance laws with nonconservative products: application to shallow flows Comput. Methods Appl. Mech. Eng. 196 2788 2815 10.1016/j.cma.2005.07.007
11. M.J. Castro T. Chacón E.D. Fernández C. Parés 2007 On well-balanced finite volume methods for non-conservative non-homogeneous hyperbolic systems SIAM J. Sci. Comput. 29 1093 1126 1148.65065 10.1137/040607642 (Pubitemid 351774662)
12. M.J. Castro A. Pardo C. Parés 2007 Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique Math. Models Methods Appl. Sci. 17 2055 2113 1137.76038 10.1142/S021820250700256X
13. M.J. Castro P.G. LeFloch M.L. Muñoz C. Parés 2008 Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes J. Comput. Phys. 3227 8107 8129 10.1016/j.jcp.2008.05. 012
14. M.J. Castro E.D. Fernández A.M. Ferreiro J.A. García C. Parés 2009 High order extensions of Roe schemes for two dimensional nonconservative hyperbolic systems J. Sci. Comput. 39 67 114 1203.65131 10.1007/s10915-008-9250-4
15. M.J. Castro A. Pardo C. Parés E.F. Toro 2010 On some fast well-balanced first order solvers for nonconservative systems Math. Comput. 79 1427 1472 05776273
16. G. Dal Maso P.G. LeFloch F. Murat 1995 Definition and weak stability of nonconservative products J. Math. Pures Appl. 74 483 548 0853.35068
17. M. Dumbser M.J. Castro C. Parés E.F. Toro 2009 ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows Comput. Fluids 38 1731 1748 1177.76222 10.1016/j.compfluid. 2009.03.008
18. Fernández-Nieto, E.D.: Aproximación numérica de leyes de conservación hiperbólicas no homogéneas. Aplicación a las ecuaciones de aguas someras. Ph.D. Thesis, Univ. Sevilla (2003)
19. E.D. Fernández F. Bouchut D. Bresh M.J. Castro A. Mangeney 2008 A new Savage-Hutter type model for submarine avalanches and generated tsunami J. Comput. Phys. 227 7720 7754 1156.76015 10.1016/j.jcp.2008.04.039
20. García Rodríguez, J.A.: Paralelización de esquemas de volúmenes finitos: aplicación a la resolución de sistemas de tipo aguas someras. Ph.D. Thesis, Univ. Málaga (2005)
21. Ishii, M.: Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris (1975)
22. A. Kurganov G. Petrova 2009 Central-upwind schemes for two-layer shallow water equations SIAM J. Sci. Comput. 31 1742 1773 1188.76230 10.1137/080719091
23. C. Parés 2006 Numerical methods for nonconservative hyperbolic systems: a theoretical framework SIAM J. Numer. Anal. 44 1 300 321 1130.65089 10.1137/050628052 (Pubitemid 46352074)
24. C. Parés M.J. Castro 2004 On the well-balance property of Roes method for nonconservative hyperbolic systems. Applications to Shallow-Water Systems Model. Math. Anal. Numer. 38 5 821 852 1130.76325 10.1051/m2an:2004041 (Pubitemid 39603844)
25. C. Parés M.L. Muñoz-Ruiz 2009 On some difficulties of the numerical approximation of nonconservative hyperbolic systems Bol. Soc. Esp. Mat. Apl. 47 23 52
26. M. Pelanti F. Bouchut A. Mangeney 2008 A Roe-type scheme for two-phase shallow granular flows over variable topography Model. Math. Anal. Numer. 42 851 885 05351740 10.1051/m2an:2008029 (Highlight paper)
27. Schijf, J.B.; Schonfeld, J.C.: Theoretical considerations on the motion of salt and fresh water. In: Proc. of the Minn. Int. Hydraulics Conv.; pp. 321-333. Joint meeting IAHR and Hyd. Div. ASCE, Sept. 1953
28. I. Toumi 1992 A weak formulation of Roe approximate Riemann solver J. Comput. Phys. 102 2 360 373 0783.65068 10.1016/0021-9991(92)90378-C
29. A.I. Volpert 1967 Spaces BV and quasilinear equations Math. USSR Sb. 73 255 302
30. J. Zabsonré G. Narbona-Reina 2009 Existence of a global weak solution for a 2D viscous bi-layer shallow water model Nonlinear Anal. Real World Appl. 5 2971 2984 10.1016/j.nonrwa.2008.09.004