Numerical solution of shallow water magnetohydrodynamic equations with non-flat bottom topography

References

• Babatin, M.M., and Y.H. Zahran. 2009. “Adaptive Multi-Resolution Central-Upwind Schemes for Systems of Conservation Laws.” International Journal of Computational Fluid Dynamics 23: 723–735.
   Web of Science ®Google Scholar
• Brackbill, J.U., and D.C. Barnes. 1980. “The Effect of Nonzero ∇.B on the Numerical Solution of the Magnetohydrodynamic Equations.” Journal of Computational Physics 35: 426–430.
   Web of Science ®Google Scholar
• Croisille, J.-P., R. Khanfir, and G. Chanteur. 1995. “Numerical Simulation of the MHD Equations by Kinetic-Type Method.” Journal of Scientific Computing 10: 81–92.
   Google Scholar
• De Sterck, H. 2001. “Hyperbolic Theory of “Shallow Water” Magnetohydrodynamics Equations.” Physics of Plasmas 8: 3293–3304.
   Web of Science ®Google Scholar
• Evans, C.R., and J.F. Hawley. 1988. “Simulation of Magnetohydrodynamic Flows – a Constrained Transport Method.” The Astrophysical Journal Letters 332: 659–677.
   Web of Science ®Google Scholar
• Gilman, P.A. 2000. “Magnetohydrodynamic “Shallow Water” Equations for the Solar Tachocline.” The Astrophysical Journal Letters 544: 79–82.
   Web of Science ®Google Scholar
• Heng, K., and A. Spitkovsky. 2009. “Magnetohydrodynamic Shallow Water Waves: Linear Analysis.” The Astrophysical Journal 703: 1819–1831.
   Web of Science ®Google Scholar
• Inogamov, N.A., and R.A. Sunyaev. 1999. “Spread of Matter over a Neutron-Star Surface During Disk Accretion.” Astronomy Letters 25: 269–293.
   Web of Science ®Google Scholar
• Jin, S. 2001. “A Steady-State Capturing Method for Hyperbolic System with Geometrical Source Terms.” Mathematical Modelling and Analysis 35: 631–646.
   Web of Science ®Google Scholar
• Kröger, T., and M. Lukáčová-Medvid’ová. 2005. “An Evolution Galerkin Scheme for the Shallow Water Magnetohydrodynamic Equations in Two Space Dimensions.” Journal of Computational Physics 206: 122–149.
   Web of Science ®Google Scholar
• Kunik, M., S. Qamar, and G. Warnecke. 2005. “A BGK-Type Kinetic Flux-Vector Splitting Schemes for the Ultra-relativistic Gas Dynamics.” Journal of Scientific Computing 205: 182–204.
   Google Scholar
• Kurganov, A., and G. Petrova. 2007. “A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System.” Communications in Mathematical Sciences 5: 133–160.
   Web of Science ®Google Scholar
• Kurganov, A., and E. Tadmor. 2000. “New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection Diffusion Equations.” Journal of Computational Physics. 160: 241–282.
   Web of Science ®Google Scholar
• LeVeque, R.J. 1998. “Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasisteady Wave-Propagation Algorithm.” Journal of Computational Physics 146: 346–365.
   Web of Science ®Google Scholar
• Lukáčová-Medvid’ová, M., and J. Saibertová. 2003. “Genuinely Multidimensional Evolution Galerkin Scheme for the Shallow Water Equations.” Enumath Conference in: Numerical Mathematics and Advanced Applications, F. Brezzi et al. (Eds.), World Scientific Publishing Company, Singapore, pp. 105–114.
   Google Scholar
• Mandal, J.C., and S.M. Deshpande. 1994. “Kinetic Flux-Vector Splitting for Euler Equations.” Computers & Fluids 23: 447–478.
   Web of Science ®Google Scholar
• Noelle, S., N. Pankratz, G. Puppo, and J. Natvig. 2006. “Well-Balanced Finite Volume Schemes of Arbitrary Order of Accuracy for Shallow Water Flows.” Journal of Computational Physics 213: 474–499.
   Web of Science ®Google Scholar
• Orszag, S.A., and M. Tang. 1979. “Small-Scale Structure of Two-Dimensional Magnetohydrodynamic Turbulence.” Journal of Fluid Mechanics 90: 129–143.
   Web of Science ®Google Scholar
• Powell, K.G. 1994. “Approximate Riemann Solver for Magnetohydrodynamics (That Works in More Than One Dimension).” Technical Report ICASE Report 94-24, ICASE, NASA Langley.
   Google Scholar
• Powell, K.G., P.L. Roe, T.J. Linde, T.I. Gombosi, and D.L. de Zeeuw. 1999. “A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics.” Journal of Computational Physics 154: 284–309.
   Web of Science ®Google Scholar
• Qamar, S., and S. Mudasser. 2010. “A Kinetic Flux-Vector Splitting Method for the Shallow Water Magnetohydrodynamics.” Computer Physics Communications 181: 1109–1122.
   Web of Science ®Google Scholar
• Rossmanith, J.A. 2002. “A Wave Propagation Method with Constrained Transport for Ideal and Shallow Water Magnetohydrodynamics.” PhD diss., University of Washington, Seattle, Washington.
   Google Scholar
• Schecter, D.A., J.F. Boyd, and G. Gilman. 2001. “Shallow-Water Magnetohydrodynamic Waves in the Solar Tachocline.” The Astrophysical Journal Letters 551: L185–L188.
   Web of Science ®Google Scholar
• Spiegel, E.A., and J.-P. Zahn. 1992. “The Solar Tachocline.” Astronomy & Astrophysics 265: 106–114.
   Web of Science ®Google Scholar
• Tang, H., T. Tang, and K. Xu. 2004. “A Gas-Kinetic Scheme for Shallow-Water Equations with Source Terms.” Zeitschrift für Angewandte Mathematik und Physik 55: 365–382.
   Web of Science ®Google Scholar
• Toro, E.F. 2001. Shock-Capturing Methods for Free Surface Shallow Flows. Vol. 2. New York: Wiley.
   Google Scholar
• Tóth, G. 2000. “The ∇.B Constraint in Shock-Capturing Magnetohydrodynamics Codes.” Journal of Computational Physics 161: 605–652.
   Web of Science ®Google Scholar
• Xing, Y., and C.-W. Shu. 2005. “High Order Finite Difference WENO Schemes with the Exact Conservation Property for the Shallow Water Equations.” Journal of Computational Physics 208: 206–227.
   Web of Science ®Google Scholar
• Xu, K. 1999. “Gas-Kinetic Theory Based Flux Splitting Method for Ideal Magnetohydrodynamics.” Journal of Computational Physics 153: 334–352.
   Web of Science ®Google Scholar
• Xu, K. 2002. “A Well-Balanced Gas-Kinetic Scheme for the Shallow-Water Equations with Source Terms.” Journal of Computational Physics 178: 533–562.
   Web of Science ®Google Scholar
• Zahran, Y.H. 2007. “Non-oscillatory Central-Upwind Scheme for Hyperbolic Conservation Laws.” International Journal of Computational Fluid Dynamics 21: 11–19.
   Web of Science ®Google Scholar
• Zaqarashvili, T.V., R. Oliver, and J.L. Ballester. 2009. “Global Shallow Water Magnetohydrodynamic Waves in the Solar Tachocline.” The Astrophysical Journal Letters 691: L41–L44.
   Web of Science ®Google Scholar
• Zaqarashvili, T.V., R. Oliver, J.L. Ballester, and B.M. Shergelashvili. 2007. “Rossby Waves in “Shallow Water” Magnetohydrodynamics.” Astronomy & Astrophysics 470: 815–820.
   Web of Science ®Google Scholar