Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SPLITTING TECHNIQUE AND GODUNOV-TYPE SCHEMES FOR 2D SHALLOW WATER EQUATIONS WITH VARIABLE TOPOGRAPHY

  • Dao Huy Cuong (Department of Mathematics Ho Chi Minh City University of Education) ;
  • Mai Duc Thanh (Department of Mathematics International University and Vietnam National University)
  • Received : 2023.09.21
  • Accepted : 2024.03.25
  • Published : 2024.07.31
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Abstract

We present numerical schemes to deal with nonconservative terms in the two-dimensional shallow water equations with variable topography. Relying on the dimensional splitting technique, we construct Godunov-type schemes. Such schemes can be categorized into two classes, namely the partly and fully splitting ones, depending on how deeply the scheme employs the splitting method. An upwind scheme technique is employed for the evolution of the velocity component for the partly splitting scheme. These schemes are shown to possess interesting properties: They can preserve the positivity of the water height, and they are well-balanced.

Keywords

Acknowledgement

We are very grateful to the reviewer for his/her very constructive comments and helpful suggestions. This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2023-28-07.

References

  1. A. Ambroso, C. Chalons, F. Coquel, and T. Gali'e, Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, M2AN Math. Model. Numer. Anal. 43 (2009), no. 6, 1063-1097. https://doi.org/10.1051/m2an/2009038 
  2. A. Ambroso, C. Chalons, P.-A. Raviart, A Godunov-type method for the seven-equation model of compressible two-phase flow, Computers and Fluids, 54 (2012), 67-91.  https://doi.org/10.1016/j.compfluid.2011.10.004
  3. E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput. 25 (2004), no. 6, 2050-2065. https://doi.org/10.1137/S1064827503431090 
  4. M. Baudin, F. Coquel, and Q.-H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline, SIAM J. Sci. Comput. 27 (2005), no. 3, 914-936. https://doi.org/10.1137/030601624 
  5. R. Botchorishvili, B. Perthame, and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comp. 72 (2003), no. 241, 131-157. https://doi.org/10.1090/S0025-5718-01-01371-0 
  6. R. Botchorishvili and O. Pironneau, Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws, J. Comput. Phys. 187 (2003), no. 2, 391-427. https://doi.org/10.1016/S0021-9991(03)00086-X 
  7. A. Chinnayya, A.-Y. LeRoux, and N. Seguin, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon, Int. J. Finite Vol. 1 (2004), no. 1, 33 pp. 
  8. F. Coquel, J.-M. Herard, K. Saleh, and N. Seguin, Two properties of two-velocity two-pressure models for two-phase flows, Commun. Math. Sci. 12 (2014), no. 3, 593-600. https://doi.org/10.4310/CMS.2014.v12.n3.a10 
  9. D. H. Cuong and M. D. Thanh, A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section, Appl. Math. Comput. 256 (2015), 602-629. https://doi.org/10.1016/j.amc.2015.01.024 
  10. D. H. Cuong and M. D. Thanh, A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography, Adv. Comput. Math. 43 (2017), no. 5, 1197-1225. https://doi.org/10.1007/s10444-017-9521-4 
  11. D. H. Cuong and M. D. Thanh, A high-resolution van Leer-type scheme for a model of fluid flows in a nozzle with variable cross-section, J. Korean Math. Soc. 54 (2017), no. 1, 141-175. https://doi.org/10.4134/JKMS.j150616 
  12. G. Dal Maso, P. G. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483-548. 
  13. U. S. Fjordholm, S. Mishra, and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, J. Comput. Phys. 230 (2011), no. 14, 5587-5609. https://doi.org/10.1016/j.jcp.2011.03.042 
  14. J. M. Gallardo, C. Par'es, and M. Castro, On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys. 227 (2007), no. 1, 574-601. https://doi.org/10.1016/j.jcp.2007.08.007 
  15. T. Gallouet, J.-M. Herard, N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. & Fluids, 32 (2003), 479-513.  https://doi.org/10.1016/S0045-7930(02)00011-7
  16. P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincar'e C Anal. Non Lineaire 21 (2004), no. 6, 881-902. https://doi.org/10.1016/j.anihpc.2004.02.002 
  17. J. M. Greenberg and A. Y. LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 1-16. https://doi.org/10.1137/0733001 
  18. T. Y. Hou and P. G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), no. 206, 497-530. https://doi.org/10.2307/2153520 
  19. E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math. 52 (1992), no. 5, 1260-1278. https://doi.org/10.1137/0152073 
  20. E. Isaacson and B. Temple, Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math. 55 (1995), no. 3, 625-640. https://doi.org/10.1137/S0036139992240711 
  21. B. L. Keyfitz, R. Sanders, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 4, 541-563. https://doi.org/10.3934/dcdsb.2003.3.541 
  22. P. G. LeFloch and M. D. Thanh, The Riemann problem for the shallow water equations with discontinuous topography, Commun. Math. Sci. 5 (2007), no. 4, 865-885. http://projecteuclid.org/euclid.cms/1199377555  https://doi.org/10.4310/CMS.2007.v5.n4.a7
  23. P. G. LeFloch and M. D. Thanh, A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime, J. Comput. Phys. 230 (2011), no. 20, 7631-7660. https://doi.org/10.1016/j.jcp.2011.06.017 
  24. D. Marchesin and P. J. Paes-Leme, A Riemann problem in gas dynamics with bifurcation, Comput. Math. Appl. Part A 12 (1986), no. 4-5, 433-455.  https://doi.org/10.1016/0898-1221(86)90173-2
  25. M. Ricchiuto and A. Bollermann, Stabilized residual distribution for shallow water simulations, J. Comput. Phys. 228 (2009), no. 4, 1071-1115. https://doi.org/10.1016/j.jcp.2008.10.020 
  26. R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), no. 2, 425-467. https://doi.org/10.1006/jcph.1999.6187 
  27. D. W. Schwendeman, C. W. Wahle, and A. K. Kapila, The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J. Comput. Phys. 212 (2006), no. 2, 490-526. https://doi.org/10.1016/j.jcp.2005.07.012 
  28. M. D. Thanh, The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math. 69 (2009), no. 6, 1501-1519. https://doi.org/10.1137/080724095 
  29. M. D. Thanh, A phase decomposition approach and the Riemann problem for a model of two-phase flows, J. Math. Anal. Appl. 418 (2014), no. 2, 569-594. https://doi.org/10.1016/j.jmaa.2014.04.012 
  30. M. D. Thanh, D. H. Cuong, and D. X. Vinh, The resonant cases and the Riemann problem for a model of two-phase flows, J. Math. Anal. Appl. 494 (2021), no. 1, Paper No. 124578, 28 pp. https://doi.org/10.1016/j.jmaa.2020.124578 
  31. N. X. Thanh, M. D. Thanh, and D. H. Cuong, Dimensional splitting well-balanced schemes on Cartesian mesh for 2D shallow water equations with variable topography, Bull. Iranian Math. Soc. 48 (2022), no. 5, 2321-2348. https://doi.org/10.1007/s41980-021-00648-x 
  32. B. Tian, E. F. Toro, and C. E. Castro, A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver, Comput. & Fluids 46 (2011), 122-132. https://doi.org/10.1016/j.compfluid.2011.01.038