A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis. (English) Zbl 1481.35331
Summary: The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35C08 | Soliton solutions |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
initial-boundary value problem; Galerkin/finite element method; BBM-BBM system; solitary waves; slip-wall boundary conditions; regularized shallow water equationsSoftware:
HE-E1GODFReferences:
| [1] | Adamy, K., Existence of solutions for a Boussinesq system on the half line and on a finite interval, Discrete Contin. Dyn. Syst. A, 29, 25-49 (2011) · Zbl 1218.35004 · doi:10.3934/dcds.2011.29.25 |
| [2] | Ali, A.; Kalisch, H., Mechanical balance laws for Boussinesq models of surface water waves, J. Nonlinear Sci., 22, 371-398 (2012) · Zbl 1253.35113 · doi:10.1007/s00332-011-9121-2 |
| [3] | Antonopoulos, D.; Dougalis, V.; Mitsotakis, D., Initial-boundary value problems for the Bona-Smith family of Boussinesq systems, Adv. Differ. Equ., 14, 27-53 (2009) · Zbl 1171.35457 |
| [4] | Antonopoulos, D.; Dougalis, V.; Mitsotakis, D., Numerical solution of Boussinesq systems of the Bona-Smith family, Appl. Num. Math., 60, 314-336 (2010) · Zbl 1303.76082 · doi:10.1016/j.apnum.2009.03.002 |
| [5] | Behzadan, A.; Holst, M., Multiplication in Sobolev spaces, revisited (2015) · Zbl 1490.46027 |
| [6] | Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. A, 272, 47-78 (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032 |
| [7] | Berger, M. S., Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis (1977), New York: Academic, New York · Zbl 0368.47001 |
| [8] | Bona, J. L.; Chen, M., A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D, 116, 191-224 (1998) · Zbl 0962.76515 · doi:10.1016/s0167-2789(97)00249-2 |
| [9] | Bona, J.; Chen, M.; Saut, J-C, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and linear theory, J. Nonlinear Sci., 12, 283-318 (2002) · Zbl 1022.35044 · doi:10.1007/s00332-002-0466-4 |
| [10] | Bona, J. L.; Colin, T.; Lannes, D., Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178, 373-410 (2005) · Zbl 1108.76012 · doi:10.1007/s00205-005-0378-1 |
| [11] | Bona, J. L.; Dougalis, V. A.; Mitsotakis, D. E., Numerical solution of KdV-KdV systems of Boussinesq equations, Math. Comput. Simul., 74, 214-228 (2007) · Zbl 1158.35410 · doi:10.1016/j.matcom.2006.10.004 |
| [12] | Bona, J. L.; Dougalis, V. A.; Mitsotakis, D. E., Numerical solution of Boussinesq systems of KdV-KdV type: II. Evolution of radiating solitary waves, Nonlinearity, 21, 2825-2848 (2008) · Zbl 1158.35411 · doi:10.1088/0951-7715/21/12/006 |
| [13] | Bona, J. L.; Smith, R., A model for the two-way propagation of water waves in a channel, Math. Proc. Camb. Phil. Soc., 79, 167-182 (1976) · Zbl 0332.76007 · doi:10.1017/s030500410005218x |
| [14] | Brenner, S.; Scott, R., The Mathematical Theory of Finite Element Methods (2007), Berlin: Springer, Berlin |
| [15] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), Berlin: Springer, Berlin · Zbl 1220.46002 |
| [16] | Chen, M., Equations for bi-directional waves over an uneven bottom, Math. Comput. Simul., 62, 3-9 (2003) · Zbl 1013.76014 · doi:10.1016/s0378-4754(02)00193-3 |
| [17] | Chubarov, L. B.; Shokin, Y. I., The numerical modelling of long wave propagation in the framework of nonlinear dispersion models, Comput. Fluids, 15, 229-249 (1987) · Zbl 0638.76014 · doi:10.1016/0045-7930(87)90008-9 |
| [18] | Clamond, D.; Dutykh, D., Practical use of variational principles for modeling water waves, Physica D, 241, 25-36 (2012) · Zbl 1431.76030 · doi:10.1016/j.physd.2011.09.015 |
| [19] | Dougalis, V. A.; Mitsotakis, D. E.; Saut, J-C, On some Boussinesq systems in two space dimensions: theory and numerical analysis, ESAIM: Math. Modelling Numer. Anal., 41, 825-854 (2007) · Zbl 1140.76314 · doi:10.1051/m2an:2007043 |
| [20] | Dougalis, V. A.; Mitsotakis, D.; Mitsotakis, D. E.; Saut, J-C, On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain, Discrete Contin. Dyn. Syst., 23, 1191-1204 (2009) · Zbl 1155.35431 · doi:10.3934/dcds.2009.23.1191 |
| [21] | Dougalis, V. A.; Mitsotakis, D. E.; Saut, J-C, Boussinesq systems of Bona-Smith type on plane domains: theory and numerical analysis, J. Sci. Comput., 44, 109-135 (2010) · Zbl 1203.65179 · doi:10.1007/s10915-010-9368-z |
| [22] | Duchêne, V.; Israwi, S., Well-posedness of the Green-Naghdi and Boussinesq-Peregrine systems, Ann. Math. Blaise Pasca, 25, 21-74 (2018) · Zbl 1405.35159 · doi:10.5802/ambp.372 |
| [23] | Dutykh, D.; Mitsotakis, D.; Chubarov, L. B.; Shokin, Y. I., On the contribution of the horizontal sea-bed displacements into the tsunami generation process, Ocean Model., 56, 43-56 (2012) · doi:10.1016/j.ocemod.2012.07.002 |
| [24] | Dutykh, D.; Mitsotakis, D.; Gardeil, X.; Dias, F., On the use of the finite fault solution for tsunami generation problems, Theor. Comput. Fluid Dyn., 27, 177-199 (2013) · doi:10.1007/s00162-011-0252-8 |
| [25] | Fokas, A. S.; Pelloni, B., Boundary value problems for Boussinesq type systems, Math. Phys. Anal. Geom., 8, 59-96 (2005) · Zbl 1070.35030 · doi:10.1007/s11040-004-1650-6 |
| [26] | Girault, V.; Raviart, P-A, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol 5 (1986), Berlin: Springer, Berlin · Zbl 0585.65077 |
| [27] | Grisvard, P., Quelques proprietés des espaces de Sobolev utiles dans l’ étude des équations de Navier-Stokes (i) (19741976) |
| [28] | Haroske, D.; Triebel, T., Distributions, Sobolev Spaces, Elliptic Equations (2007), Zurich: European Mathematical Society, Zurich |
| [29] | Katsaounis, T.; Mitsotakis, D.; Sadaka, G., Boussinesq-Peregrine water wave models and their numerical approximation, J. Comput. Phys., 417 (2020) · Zbl 1437.76022 · doi:10.1016/j.jcp.2020.109579 |
| [30] | Khakimzyanov, G.; Dutykh, D., Long wave interaction with a partially immersed body: I. Mathematical models, Commun. Comp. Phys., 27, 321-378 (2020) · Zbl 1490.76039 · doi:10.4208/cicp.OA-2018-0294 |
| [31] | Lannes, D., The Water Waves Problem: Mathematical Analysis and Asymptotics, vol 188 (2013), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1410.35003 |
| [32] | Lannes, D.; Bonneton, P., Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Phys. Fluids, 21 (2009) · Zbl 1183.76294 · doi:10.1063/1.3053183 |
| [33] | LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1010.65040 |
| [34] | Mitsotakis, D. E., Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves, Math. Comput. Simul., 80, 860-873 (2009) · Zbl 1256.35089 · doi:10.1016/j.matcom.2009.08.029 |
| [35] | Mitsotakis, D.; Ranocha, H.; Ketcheson, D. I.; Süli, E., A conservative fully discrete numerical method for the regularized shallow water wave equations, SIAM J. Sci. Comput., 43, B508-B537 (2021) · Zbl 1468.65146 · doi:10.1137/20m1364606 |
| [36] | Monk, P., Finite Element Methods for Maxwell’s Equations (2003), Oxford: Oxford University Press, Oxford · Zbl 1024.78009 |
| [37] | Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hambg., 36, 9-15 (1971) · Zbl 0229.65079 · doi:10.1007/bf02995904 |
| [38] | Peregrine, D. H., Calculations of the development of an undular bore, J. Fluid Mech., 25, 321-330 (1966) · doi:10.1017/s0022112066001678 |
| [39] | Peregrine, D. H., Long waves on a beach, J. Fluid Mech., 27, 815-827 (1967) · Zbl 0163.21105 · doi:10.1017/s0022112067002605 |
| [40] | Saut, J-C; Xu, L., The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl., 97, 635-662 (2012) · Zbl 1245.35090 · doi:10.1016/j.matpur.2011.09.012 |
| [41] | Scott, R., Optimal L^∞ estimates for the finite element method on irregular meshes, Math. Comp., 30, 681-697 (1976) · Zbl 0349.65060 · doi:10.1090/s0025-5718-1976-0436617-2 |
| [42] | Seliger, R. L.; Whitham, G. B., Variational principles in continuum mechanics, Proc. R. Soc. A, 305, 1-25 (1968) · Zbl 0198.57601 · doi:10.1098/rspa.1968.0103 |
| [43] | Senthikumar, A., On the influence of wave reflection on shoaling and breaking solitary waves, Proc. Est. Acad. Sci., 65, 414-430 (2016) · Zbl 1386.76044 · doi:10.3176/proc.2016.4.06 |
| [44] | Tao, T., Local and Global Analysis of Nonlinear Dispersive and Wave Equations (2006), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1106.35001 |
| [45] | Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (2006), Berlin: Springer, Berlin · Zbl 1105.65102 |
| [46] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2013), Berlin: Springer, Berlin |
| [47] | Walkley, M.; Berzins, M., A finite element method for the one-dimensional extended Boussinesq equations, Int. J. Numer. Methods Fluids, 29, 143-157 (1999) · Zbl 0941.76055 · doi:10.1002/(sici)1097-0363(19990130)29:2<143::aid-fld779>3.0.co;2-5 |
| [48] | Whitham, G. B., Linear and Nonlinear Waves (2011), New York: Wiley, New York |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
