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Numerical Methods for a Class of Shallow Water Equation(PDF)

《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

Issue:
2015年01期
Page:
32-
Research Field:
数学
Publishing date:

Info

Title:
Numerical Methods for a Class of Shallow Water Equation
Author(s):
Zhang Jun1; Fan Xinyue2
(1.School of Mathematics and Statistics,Guizhou University of Finance and Economics,Guiyang 550025,China)(2.College of Science,Guizhou University,Guiyang 550025,China)
Keywords:
BBM equation; unconditionally stable; finite difference method; spectral method; decay rate
PACS:
O156.5
DOI:
-
Abstract:
We turn to study the numerical solution of the shallow water equation. We propose two different schemes to numerically solve this equation. A detailed analysis is carried out for these schemes,and we prove that the overall schemes are unconditionally stable. The error estimation shows that the linearized Euler schema in time plus Fourier spectral method in space is convergent with the convergence order O(Δt+N1-m),and higher order convergences can be obtained if the second order backward differentiation or Crank-Nicolson schema are used to discretize the equation in time. At last,we use the proposed methods to investigate the asymptotical decay rate of the solutions to the shallow water wave equation. We equally discuss the role of the diffusion terms,the geometric dispersion and the nonlinearity respectively. The performed numerical experiment confirms that the decay rates in L2-norm,L∞-norm,and H1-seminorm are very close to -1/4,-1/2,and -3/4 respectively. These numerical results are consistent with the known theoretical prediction.

References:

[1] Russell J. Report on waves[C]//14th Meeting of the British Association for the Advancement of Science.London,1844:311-390.
[2]Korteweg D D J,deVries D G. On the change of form of long waves advancing in a rectangular canal,and on a new type of long stationary waves[J]. The London,Edinburgh,and Dublin Philosophical Magazine and Journal of Science,1895,39(240):422-443.
[3]Benjamin T B,Bona J L,Mahony J J. Model equations for long waves in nonlinear dispersive systems[J]. Philosophical Trans Actions of the Royal Society of London. Series A,Mathematical and Physical Sciences,1972,272(1 220):47-78.
[4]Medeiros L A,Menzala G P. Existence and uniqueness for periodic solutions of the benjamin-bona-mahony equation[J]. SIAM Journal on Mathematical Analysis,1977,8(5):792-799.
[5]Albert J. Dispersion of low-energy waves for the generalized benjamin-bona-mahony equation[J]. Journal of Differential Equations,1986,63(1):117-134.
[6]Albert J. On the decay of solutions of the generalized benjamin-bona-mahony equation[J]. Journal of Mathematical Analysis and Applications,1989,141(2):527-537.
[7]Biler P. Long time behavior of solutions of the generalized Benjamin-Bona-Mahony equation in two space dimensions[D]. Paris:Departement de Mathematiques,Universite de Paris-sud,1991.
[8]Amick C,Bona J,Schonbek M E. Decay of solutions of some nonlinear wave equations[J]. Journal of Differential Equations,1989,81(1):1-49.
[9]Mei M. lq-decay rates of solutions for benjamin-bona-mahony-burgers equations[J]. Journal of Differential Equations,1999,158(2):314-340.
[10]Chen M,Dumont S,Dupaigne L,et al. Decay of solutions to a water wave model with a nonlocal viscous dispersive term[J]. Discrete Cont Dyn Syst-Ser A,2010,27(4):1 473-1 492.
[11]Zhang L. Decay of solutions of generalized benjamin-bona-mahony equations in n-space dimensions[J]. Nonlinear Analysis:Theory,Methods & Applications,1995,25:1 345-1 369.
[12]Fang S,Guo B. The decay rates of solutions of generalized benjamin-bona-mahony equations in multi-dimensions[J]. Nonlinear Analysis:Theory,Methods & Applications,2008,69(7):2 230-2 235.
[13]Dogan A. Numerical solution of rlw equation using linear finite elements withi galerkin’s method[J]. Applied Mathematical Modelling,2002,26(7):771-783.
[14]Saka B,Dag I,Dogan A. Galerkin method for the numerical solution of the rlw equation using quadratic b-splines[J]. International Journal of Computer Mathematics,2004,81(6):727-739.
[15]Dag I,Saka B,Irk D. Galerkin method for the numerical solution of the rlw equation using quintic b-splines[J]. Journal of Computational and Applied Mathematics,2006,190(1):532-547.
[16]Omrani K. The convergence of fully discrete galerkin approximations for the Benjamin-bona-mahony(bbm)equation[J]. Applied Mathematics and Computation,2006,180(2):614-621.
[17]Omrani K,Ayadi M. Finite difference discretization of the benjaminbona-mahony-burgers equation[J]. Numerical Methods for Partial Differential Equations,2008,24(1):239-248.
[18]Chen M. Numerical investigation of a two-dimensional boussinesq system[J]. Discrete Contin Dyn Syst,2009,28(4):1 169-1 190.
[19]Quarteroni A,Valli A. Numerical Approximation of Partial Differential Equations[M]. Berlin:Springer-Verlag,1994.
[20]Khismatullin D,Renardy Y,Renardy M. Development and implementation of vof-prost for 3d viscoelastic liquid simulations[J]. Journal of Non-Newtonian Uid Mechanics,2006,140(1):120-131.
[21]Chen M. Exact traveling-wave solutions to bi-directional wave equations[J]. International Journal of Theoretical Physics,1998,37(5):1 547-1 567.

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Last Update: 2015-03-30