Skip to main content
SpringerLink
Log in

A space–time CESE scheme for shallow water magnetohydrodynamics equations with variable bottom topography

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

The dynamics of thin layer incompressible and electrically conducting fluids is described by shallow water magnetohydrodynamics equations (SWMHD), for which the evolution is nearly two dimensional and the magnetic equilibrium lies in the third direction. In this article, a space–time conservation element and solution element (CESE) scheme is extended for numerical investigation of one- and two-dimension shallow water magnetohydrodynamics (SWMHD) with variable bottom topography. The presence of non-conservative terms and the inclusion of variable bottom topography on the right hand side of SWMHD equations poses some major challenges for any numerical scheme. The proposed CESE scheme differs from the previous techniques because of global and local flux conservation in a space–time domain without restoring to interpolation and extrapolation. To check the validity of the scheme, a number of test problems are considered. The results are also compared with the already existing results in the literature by central-upwind scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Inogamov NA, Sunyaev RA (1999) Spread of matter over a neutron-star surface during disk accretion. Astron Lett 25:269–293

    Google Scholar 

  2. Gilman PA (2000) Magnetohydrodynamic shallow water equations for the solar tachocline. Astrophys. J Lett 544:79–82

    Article  Google Scholar 

  3. Spiegel EA, Zahn J-P (1992) The solar tachocline. Astron Astrophys 265:106–114

    Google Scholar 

  4. De Sterck H (2001) Hyperbolic theory of shallow water magnetohydrodynamics equations. Phys Plasmas 8:3293–3304

    Article  MathSciNet  Google Scholar 

  5. Heng K, Spitkovsky A (2009) Magnetohydrodynamic shallow water waves. Linear Anal ApJ 703:1819

    Google Scholar 

  6. Schecter DA, Boyd JF, Gilman G (2001) Shallow-water magnetohydrodynamic waves in the solar tachocline. ApJ 551:L185

    Article  Google Scholar 

  7. Zaqarashvili TV, Oliver R, Ballester JL, Shergelashvili BM (2007) Rossby waves in “shallow water” magnetohydrodynamics. A&A 470:815–820

    Article  MATH  Google Scholar 

  8. Zaqarashvili TV, Oliver R, Ballester JL (2009) Global shallow water magnetohydrodynamic waves in the solar tachocline. ApJ 691:L41–L44

    Article  Google Scholar 

  9. Toro EF (1962) Shock-capturing methods for free surface shallow fluids. Wiley, New York, vol 2

  10. Zia S, Ahmed M, Qamar S (2014) Numerical solution of shallow water magnetohydrodynamic equations with non-flat bottom topography. Int J Comput Fluid Dyn 96:136–151

    Article  MathSciNet  Google Scholar 

  11. Qamar S, Zia S, Ashraf W (2014) The space time CE/SE method for solving single and two-phase shallow flow models. Comput Fluids 28:56–75

  12. Powell KG (1994) Approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report

  13. Powell KG, Roe PL, Linde TJ, Gombosi TI, de Zeeuw DL (1999) A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J Comput Phys 154:284–309

    Article  MathSciNet  MATH  Google Scholar 

  14. Evans CR, Hawley JF (1988) Simulation of magnetohydrodynamic flows a constrained transport method. ApJ 332:659–677

    Article  Google Scholar 

  15. Brackbill JU, Baranes DC (1980) The effect of nonzero \(\nabla.\mathbf{B}\) on the numerical solution of the magnetohydrodynamic equations. J Comput Phys 35:426–430

    Article  MathSciNet  Google Scholar 

  16. Qamar S, Mudasser S (2010) A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics. Comp Phys Commun 181:1109–1122

    Article  MathSciNet  MATH  Google Scholar 

  17. Xu K (1999) Gas-kinetic theory based flux splitting method for ideal magnetohydrodynamics. J Comput Phys 153:334–352

    Article  MathSciNet  MATH  Google Scholar 

  18. Croisille J-P, Khanfir R, Chanteur G (1995) Numerical simulation of the MHD equations by kinetic-type method. J Sci Comput 10:81–92

    Article  MATH  Google Scholar 

  19. Kröger T, Lukáčová-Medvid’ová M (2005) An evolution Galerkin scheme for the shallow water magnetohydrodynamic (SWMHD) equations in two space dimensions. J Comput Phys 206:122–149

    Article  MathSciNet  MATH  Google Scholar 

  20. Tang H, Tang T, Xu K (2004) A gas-kinetic scheme for shallow-water equations with source terms. Z Angew Math Phys 55:365–382

    Article  MathSciNet  MATH  Google Scholar 

  21. Xu K (2002) A well-balanced gas-kinetic scheme for the shallow-water equations with source terms. J Comput Phys 178:533–562

    Article  MathSciNet  MATH  Google Scholar 

  22. Kurganov A, Tadmor E (2000) New high-resolution central schemes for nonlinear conservation laws and convection diffusion equations. J Comput Phys 160:241–282

    Article  MathSciNet  MATH  Google Scholar 

  23. Rossmanith JA (2002) A wave propagation method with constrained transport for ideal and shallow water magnetohydrodynamics, Ph.D. thesis, University of Washington, Seattle, Washington

  24. Tóth G (2000) The \(\nabla.\mathbf{B}\) constraint in shock-capturing magnetohydrodynamics codes. J Comput Phys 161:605–652

    Article  MathSciNet  MATH  Google Scholar 

  25. Chang SC (1995) The method of space time conservation element and solution element—a new approach for solving the Navier Stokes and Euler equations. J Comput Phys 119:234–295

    Article  MathSciNet  Google Scholar 

  26. Chang SC, Wang XY, Chow CY (1994) New developments in the method of space-time conservation element and solution element-Applications to two-dimensional time-marching problems. NASA TM 106758.

  27. Chang SC, Wang XY, To WM (2000) Application of the space-time conservation element and solution element method to one-dimensional convection-diffusion Problems. J Comput Phys 165:189–215

    Article  MATH  Google Scholar 

  28. Chang SC, Wang XY, Chow CY (1999) The space-time conservation element and solution element method. A new high resolution and genuinely multidimensional paradigm for solving conservation laws. J Comput Phys 156:89–136

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang ZC, Yu ST, Chang SC (2002) A Space-time conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexahedeal meshes. J Comput Phys 175:168–199

    Article  MATH  Google Scholar 

  30. Qamar S, Warnecke G (2005) Application of space-time CE/SE method to shallow water magnetohydrodynamic squations. J Comput Appl Math 196:132–149

    Article  MATH  Google Scholar 

  31. Dumbser M, Castro M, Parés C, Toro EF (2009) ADER schemes on unstructured meshes for nonconservative hyperbolic systems, Applications to geophysical flows. Comput Fluids 38:1731–1748

  32. Pelanti M, Bouchut F, Mangeney A (2008) A Roe-type scheme for two-phase shallow granular flows over variable topography. ESAIM-Math Model Numer 42:851–885

    Article  MathSciNet  MATH  Google Scholar 

  33. LeVeque RJ (1998) Balancing source terms and flux gradients in high-resolution Godunov methods: the quasisteady wave-propagation algorithm. J Comput Phys 146:346–365

    Article  MathSciNet  MATH  Google Scholar 

  34. Noelle S, Pankratz N, Puppo G, Natvig J (2006) Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J Comput Phys 213:474–499. 40

  35. Xing Y, Shu CW (2005) High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J Comp Phys 208:206–227. 41

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

    M. Rehan Saleem, Saqib Zia & Shamsul Qamar

Authors
  1. M. Rehan Saleem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saqib Zia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shamsul Qamar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saqib Zia.

Additional information

Technical Editor: Marcio S. Carvalho.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saleem, M.R., Zia, S. & Qamar, S. A space–time CESE scheme for shallow water magnetohydrodynamics equations with variable bottom topography. J Braz. Soc. Mech. Sci. Eng. 39, 1563–1573 (2017). https://doi.org/10.1007/s40430-016-0678-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-016-0678-4

Keywords

Search

Navigation