Abstract
In this paper we study a Lax-Wendroff-type time discretization procedure for the finite difference weighted essentially non-oscillatory (WENO) schemes to solve one-dimensional and two-dimensional shallow water equations with source terms. In order to maintain genuinely high order accuracy and suit to problems with a rapidly varying bottom topography we use WENO reconstruction not only to the flux but also to the source terms of algebraical modified shallow water equations. Extensive simulations are performed, as a result, the WENO schemes with Lax-Wendroff-type time discretization can maintain nonoscillatory properties and more cost effective than that with Runge-Kutta time discretization.
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Aràndiga, F., Belda, A.M., Mulet, P.: Point-value WENO multiresolution applications to stable image compression. J. Sci. Comput. 43, 158–182 (2009)
Balsara, D.S.: Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics. J. Comput. Phys. 228, 5040–5054 (2009)
Balsara, D.S., Shu, C.W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Cai, Y., Navon, I.M.: Parallel block preconditioning techniques for the numerical simulation of the shallow water flow using finite element methods. J. Comput. Phys. 122, 39–50 (1995)
Chleffi, V., Valaini, A., Zanni, A.: Finite volume method for simulating extreme flood events in natural channels. J. Hydraul. Res. 41, 167–177 (2003)
Goutal, N., Maurel, F.: In: Proceedings of the Second Workshop on Dam-BreakWave Simulation. Technical Report HE-43/97/016/A. Electricité de France, Département Laboratoire National Hydraulique, Groupe Hydraulique Fluviale (1997)
Harten, A., Engguist, B., Osher, S., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)
Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov method: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 346, 146 (1998)
Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
Qiu, J.X., Shu, C.W.: Finite difference WENO schemes with Lax-Wendroff-type time discretizations. SIAM J. Sci. Comput. 24, 2185–2198 (2003)
Ricchiuto, M., Abgrall, R., Deconinck, H.: Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. J. Comput. Phys. 222, 287–331 (2007)
Rogers, B.D., Borthwick, Alistair G.L., Taylor, P.H.: Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys. 192, 422–451 (2003)
Shu, C.W.: Essential non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B., Johnson, C., Shu, C.W., Tadmor, E. (eds.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998). A. Quarteroni (ed.)
Vukovic, S., Sopta, L.: ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J. Comput. Phys. 179, 593–621 (2002)
Xing, Y.L., Shu, C.W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)
Zahran, Y.H.: An efficient WENO scheme for solving hyperbolic conservation laws. Appl. Math. Comput. 212, 37–50 (2009)
Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M.: The surface gradient method for the treatment of source terms in the shallow water equations. J. Comput. Phys. 168, 1–25 (2001)
Zhu, J., Qiu, J.X.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method, III: unstructured meshes. J. Sci. Comput. 39, 293–321 (2009)
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The research of C. Lu was supported by NSFC 40906048 and Science research fund of Nanjing University of Information Science & Technology 20090203. The research of J. Qiu was supported by NSFC 10931004.
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Lu, C., Qiu, J. Simulations of Shallow Water Equations with Finite Difference Lax-Wendroff Weighted Essentially Non-oscillatory Schemes. J Sci Comput 47, 281–302 (2011). https://doi.org/10.1007/s10915-010-9437-3
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DOI: https://doi.org/10.1007/s10915-010-9437-3