Linear-implicit and energy-preserving schemes for the Benjamin-type equations

References

• J.P. Albert, J.L. Bona, and J.M. Restrepo, Solitary-wave solutions of the Benjamin equation, SIAM J. Appl. Math. 59 (1999), pp. 2139–2161. doi: 10.1137/S0036139997321682
   Web of Science ®Google Scholar
• T.B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), pp. 559–592. doi: 10.1017/S002211206700103X
   Web of Science ®Google Scholar
• T.B. Benjamin, A new kind of solitary wave, J. Fluid Mech. 245 (1992), pp. 401–411. doi: 10.1017/S002211209200051X
   Web of Science ®Google Scholar
• T.B. Benjamin, Solitary and periodic waves of a new kind, Phil. Trans. R. Soc. A 354 (1996), pp. 1775–1806. doi: 10.1098/rsta.1996.0078
   Google Scholar
• J.P. Boyd and Z.J. Xu, Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions, Wave Motion 48 (2011), pp. 702–706. doi: 10.1016/j.wavemoti.2011.02.004
   Web of Science ®Google Scholar
• L. Brugnano, F. Iavernaro, and D. Trigiante, Energy-and quadratic invariants-preserving integrators based upon Gauss collocation formulae, SIAM J. Numer. Anal. 50 (2012), pp. 2897–2916. doi: 10.1137/110856617
   Web of Science ®Google Scholar
• W. Cai, C. Jiang, and Y. Wang, Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions, preprint (2018). Available at arXiv:1809.02704.
   Google Scholar
• D.C. Calvo and T.R. Akylas, On interfacial gravity-capillary solitary waves of the Benjamin type and their stability, Phys. Fluids 15 (2003), pp. 1261–1270. doi: 10.1063/1.1564096
   Web of Science ®Google Scholar
• E. Celledoni, V. Grimm, R.I. McLachlan, D.I. McLaren, D. O'Neale, B. Owren, and G.R.W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, J. Comput. Phys. 231 (2012), pp. 6770–6789. doi: 10.1016/j.jcp.2012.06.022
   Web of Science ®Google Scholar
• V.A. Dougalis, A. Durán, and D.E. Mitsotakis, Numerical solution of the Benjamin equation, Wave Motion 52 (2015), pp. 194–215. doi: 10.1016/j.wavemoti.2014.10.004
   Web of Science ®Google Scholar
• V.A. Dougalis, A. Durán, and D.E. Mitsotakis, Numerical approximation of solitary waves of the Benjamin equation, Math. Comput. Simul. 127 (2016), pp. 56–79. doi: 10.1016/j.matcom.2012.07.008
   Web of Science ®Google Scholar
• V.A. Dougalis, A. Durán, and D.E. Mitsotakis, Numerical approximation to Benjamin type equations. Generation and stability of solitary waves, Wave Motion 85 (2019), pp. 34–56. doi: 10.1016/j.wavemoti.2018.11.002
   Web of Science ®Google Scholar
• C. Jiang, W. Cai, and Y. Wang, A linear-implicit and energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach, preprint (2018). Available at arXiv:1808.06854.
   Google Scholar
• H. Kalisch and J.L. Bona, Models for internal waves in deep water, Discrete Contin. Dyn. Syst. 6 (2000), pp. 1–20. doi: 10.3934/dcds.2000.6.1
   Web of Science ®Google Scholar
• H. Kalisch and J.L. Bona, Models for internal waves in deep water, Discrete Contin. Dyn. Syst. 6 (2000), pp. 1–22. doi: 10.3934/dcds.2000.6.1
   Web of Science ®Google Scholar
• K. Kinugasa, Y. Miyatake, and T. Matsuo, Structure-preserving integrators for the Benjamin-type equations, preprint (2015). Available at arXiv:1507.08359.
   Google Scholar
• F. Linares, L2 global well-posedness of the initial value problem associated to the Benjamin equation, J. Differ. Equ. 152 (1999), pp. 377–393. doi: 10.1006/jdeq.1998.3530
   Google Scholar
• T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations, J. Comput. Appl. Math. 218 (2008), pp. 506–521. doi: 10.1016/j.cam.2007.08.001
   Web of Science ®Google Scholar
• R.I. McLachlan, G.R.W. Quispel, and N. Robidoux, Geometric integration using discrete gradients, Philos. Trans. R. Soc. A 357 (1999), pp. 1021–1046. doi: 10.1098/rsta.1999.0363
   Google Scholar
• H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), pp. 1082–1091. doi: 10.1143/JPSJ.39.1082
   Google Scholar
• G.R.W. Quispel and D.I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A Math. Theor. 41 (2008), p. 045206. doi: 10.1088/1751-8113/41/4/045206
   Web of Science ®Google Scholar
• J. Shen, J. Xu, and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, preprint (2017). Available at arXiv:1710.01331.
   Google Scholar
• J. Shen, J. Xu, and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys. 353 (2018), pp. 407–416. doi: 10.1016/j.jcp.2017.10.021
   Web of Science ®Google Scholar
• Y. Song and Y. Wang, Energy-Preserving algorithms for the Benjamin equation, J. Sci. Comput. 72 (2017), pp. 605–622. doi: 10.1007/s10915-017-0371-5
   Web of Science ®Google Scholar
• V. Thomée and A.S. Vasudeva Murthy, A numerical method for the Benjamin-Ono equation, BIT Numer. Math. 38 (1998), pp. 597–611. doi: 10.1007/BF02510262
   Google Scholar
• X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys. 327 (2016), pp. 294–316. doi: 10.1016/j.jcp.2016.09.029
   Web of Science ®Google Scholar
• X. Yang and L. Ju, Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model, Comput. Methods Appl. Mech. Eng. 315 (2017), pp. 691–712. doi: 10.1016/j.cma.2016.10.041
   Web of Science ®Google Scholar
• X. Yang and L. Ju, Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model, Comput. Methods Appl. Mech. Eng. 318 (2017), pp. 1005–1029. doi: 10.1016/j.cma.2017.02.011
   Web of Science ®Google Scholar
• X. Yang, J. Zhao, and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys. 333 (2017), pp. 104–127. doi: 10.1016/j.jcp.2016.12.025
   Web of Science ®Google Scholar
• J. Zhao, X. Yang, Y. Gong, and Q. Wang, A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals, Comput. Methods Appl. Mech. Eng. 318 (2017), pp. 803–825. doi: 10.1016/j.cma.2017.01.031
   Web of Science ®Google Scholar