In this paper, we propose new finite volume element schemes to numerically solve the improved Boussinesq equation with Stokes damping. The new schemes can inherit characteristic properties of the conservation of mass and the decrease of total energy from the improved Boussinesq equation with Stokes damping. Numerical experiments illustrate that the proposed schemes are second-order accuracy in space and time.
Bogolubsky, I. (1977) Some Examples of Inelastic Soliton Interaction. Computer Physics Communications, 13, 149-155. https://doi.org/10.1016/0010-4655(77)90009-1
Dehghan, M. and Shakeri, F. (2008) Use of He’s Homotopy Perturbation Method for Solving a Partial Differential Equation Arising in Modeling of Flow in Porous Media. Journal of Porous Media, 11, 765-778. https://doi.org/10.1016/j.cnsns.2009.02.021
Biswas, A., Milovic, D. and Ranasinghe, A. (2009) Solitary Waves of Boussinesq Equation in a Power Law Media. Communications in Nonlinear Science and Numerical Simulation, 14, 3738-3742. https://doi.org/10.1016/j.cnsns.2009.02.021
Christiansen, P.L., Muto, V. and Rionero, S. (1992) Solitary Wave Solutions to a System of Boussinesq-Like Equations. Chaos, Solitons and Fractals, 2, 45-50. https://doi.org/10.1016/0960-0779(92)90046-P
Yang, Z. (1998) Existence and Non-Existence of Global Solutions to a Generalized Modification of the Improved Boussinesq Equation. Mathematical Methods in the Applied Sciences, 21, 1467-1477. https://doi.org/10.1002/(SICI)1099-1476(19981110)21:16%3C1467::AID-MMA968%3E3.0.CO; 2-K
Iskandar, L. and Jain, P.C. (1980) Numerical Solutions of the Improved Boussinesq Equation. Proceedings of the Indian Academy of Sciences-Mathematical Sciences, 89, 171-181. https://doi.org/10.1007/BF02861996
El-Zoheiry, H. (2002) Numerical Study of the Improved Boussinesq Equation. Chaos, Solitons and Fractals, 14, 377-384. https://doi.org/10.1016/S0960-0779(00)00271-X
Bratsos, A.G. (2007) A Second-Order Numerical Scheme for the Improved Boussinesq Equation. Physics Letters A, 370, 145-147. https://doi.org/10.1016/j.physleta.2007.05.050
Bratsos, A.G. (2009) A Predictor-Corrector Scheme for the Improved Boussinesq Equation. Chaos, Solitons & Fractals, 40, 2083-2094. https://doi.org/10.1016/j.chaos.2007.09.083
Lin, Q., Wu, Y., Loxton, R. and Lai, S. (2009) Linear B-Spline Finite Element Method for the Improved Boussinesq Equation. Journal of Computational and Applied Mathematics, 224, 658-667. https://doi.org/10.1016/j.cam.2008.05.049
Shokri, A. and Dehghan, M. (2010) A Not-a-Knot Meshless Method Using Radial Basis Functions and Predictor—Corrector Scheme to the Numerical Solution of Improved Boussinesq Equation. Computer Physics Communications, 181, 1990-2000. https://doi.org/10.1016/j.cpc.2010.08.035
Irk, D. and Dağ, İ. (2009) Numerical Simulations of the Improved Boussinesq Equation. Numerical Methods for Partial Differential Equations, 26, 1316-1327. https://doi.org/10.1002/num.20492
Zhang, Z. and Lu, F. (2012) Quadratic Finite Volume Element Method for the Improved Boussinesq Equation. Journal of Mathematical Physics, 53, Article ID: 013505. https://doi.org/10.1063/1.3672197
Yan, Z., Xie, F. and Zhang, H. (2001) Symmetry Reduction, Integrability and Solitary Wave Solutions to High-Order Modified Boussinesq Equation with Damping term. Communications in Theoretical Physics, 36, 1-6. https://doi.org/10.1088/0253-6102/36/1/1
Arevalo, E., Gaididei, Y. and Mertens, F.G. (2002) Soliton Dynamics in Damped and Forced Boussinesq Equations. The European Physical Journal B, 27, 63-74.
Chen, G., Rui, W. and Chen, X. (2011) Cauchy Problem for a Damped Generalized IMBq Equation. Journal of Mathematical Physics, 52, Article ID: 053504. https://doi.org/10.1063/1.3577956
Li, R., Chen, Z. and Wu, W. (1999) Generalized Difference Methods for Differential Equation: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York.
Zhang, Z. (2008) Error Estimates of Finite Volume Element Method for the Pollution in Groundwater Flow. Numerical Methods for Partial Differential Equations, 25, 259-274. https://doi.org/10.1002/num.20340
Li, J., Chen, Z. and He, Y. (2012) A Stabilized Multi-Level Method for Non-Singular Finite Volume Solutions of the Stationary 3D Navier-Stokes Equations. Numerische Mathematik, 122, 279-304. https://doi.org/10.1007/s00211-012-0462-z
Luo, Z., Li, H., Sun, P., An, J. and Navon, I. (2013) A Reduced-Order Finite Volume Element Formulation Based on POD Method and Numerical Simulation for Two-Dimensional Solute Transport Problems. Mathematics and Computers in Simulation, 89, 50-68. https://doi.org/10.1016/j.matcom.2012.11.012
Wang, Q., Zhang, Z. and Li, Z. (2013) A Fourier Finite Volume Element Method for Solving Two-Dimensional Quasi-Geostrophic Equations on a Sphere. Applied Numerical Mathematics, 71, 1-13. https://doi.org/10.1016/j.apnum.2013.03.007
Wang, Q. and Zhang, Z. (2014) High-Order Upwind Finite Volume Element Schemes for Modeling of Neuronal Firing. International Journal of Computer Mathematics, 91, 625-640. https://doi.org/10.1080/00207160.2013.801463
Lin, Z., Ricardo, R. and Tian, C. (2014) Finite Volume Element Approximation of an Inhomogeneous Brusselator Model with Cross-Diffusion. Journal of Computational Physics, 256, 806-823. https://doi.org/10.1016/j.jcp.2013.09.009
Furihata, D. (2001) A Stable and Conservative Finite Difference Scheme for the Cahn-Hilliard Equation. Numerische Mathematik, 87, 675-699. https://doi.org/10.1007/PL00005429
Matsuo, T. (2007) New Conservative Schemes with Discrete Variational Derivatives for Nonlinear Wave Equations. Journal of Computational and Applied Mathematics, 203, 32-56. https://doi.org/10.1016/j.cam.2006.03.009