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Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

Published online by Cambridge University Press:  28 May 2015

Changna Lu  and
Gang Li
Changna Lu*
Affiliation:
College of Mathematics & Physics, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, P.R. China
Gang Li*
Affiliation:
School of Mathematical Science, Qingdao University, Qingdao 266071, P. R. China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory (WENO) schemes to solve the one-dimensional and two-dimensional shallow water equations with source terms. Multilevel time discretization methods can make full use of computed information by WENO spatial discretization and save CPU cost by holding the former computational values. Extensive simulations are performed, which indicate that, the finite difference WENO schemes with multilevel time discretization can achieve higher accuracy, and are more cost effective than WENO scheme with Runge-Kutta time discretization, while still maintaining nonoscillatory properties.

Keywords

65M1078A48Multilevel time discretizationweighted essentially non-oscillatory schemesshallow water equationsRunge-Kutta methodhigh order accuracy
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 505 - 524
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Anastasiou, K., Chan, C. T., Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes, Int. J. Numer. Methods Fluids, 24(1997) 1225–1245.3.0.CO;2-D>CrossRefGoogle Scholar
[2]Arĺd’ndiga, F., Belda, A. M., Mulet, P., Point-Value WENO Multiresolution Applications to Stable Image Compression, J. Sci. Comput., 43(2009) 158–182.CrossRefGoogle Scholar
[3]Balsara, D., Shu, C. W., Monotonicity preserving weithted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160(2000) 405–452.CrossRefGoogle Scholar
[4]Balsara, D., Divergence-free reconstruction of magnetic fields and WENO schemes for mag-netohydrodynamics, J. Comput. Phys., 228(2009) 5040-5054.CrossRefGoogle Scholar
[5]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(1995) 39–50.CrossRefGoogle Scholar
[6]Chleffi, V., valaini, A., Zanni, A., Finite volume method for simulating extreme flood events in natural channels, J. Hydraul. Res., 41(2003) 167-177.Google Scholar
[7]Gerolymos, G. A., Sénéchal, D., Vallet, I., Very-high-order WENO schemes, J. Comput. Phys., 228(1999) 8481-8524.Google Scholar
[8]Harten, A., Engguist, B., Osher, S., Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes,, J. Comput. Phys., v71(1987) 231–303.CrossRefGoogle Scholar
[9]Harten, A., Osher, S., Uniformly high-order accurate non-oscillatory schemes, I, SIAM J. Num. Analy., v24 (1987), 279–309.CrossRefGoogle Scholar
[10]Hu, C., Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150(1999) 97–127.CrossRefGoogle Scholar
[11]Jiang, G. S., Shu, C. W., Effecient implementation of weighted ENO schemes, J. Comput. Phys., 126(1996) 202–228.CrossRefGoogle Scholar
[12]LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov method: the quasi-steady wave-propagation algorithm, J. Comput. Phys., 346(1998) 146.Google Scholar
[13]Liu, X. D., Osher, S., Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115(1994) 200–212.CrossRefGoogle Scholar
[14]Qiu, J. X., Shu, C. W., Finite difference WENO schemes with Lax-wendroff-type time discretizations, SIAM J. Sci. Comput., 24(2003) 2185-2198.CrossRefGoogle Scholar
[15]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(2007) 287–331.CrossRefGoogle Scholar
[16]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(2003) 422–451.CrossRefGoogle Scholar
[17]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.. Advanced numerical approximation of nonlinear hyperbolic equations. In: A Quarteroni, editor. Lecture Notes in Mathematics, Vol. 1697. Berlin: Springer, 1998. 325–432.CrossRefGoogle Scholar
[18]Shu, C. W., Total-variation-diminishing time discretizations, SIAM J. Sci. and Stat. Comput., 24(2003) 2185-2198.Google Scholar
[19]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(2005) 206-227.CrossRefGoogle Scholar
[20]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(2009) 293–321.Google Scholar
[21]Zahran, Y. H., An efficient WENO scheme for solving hyperbolic conservation laws, Appl. Math. Comput., 212(2009) 37–50.Google Scholar