Rectification of a deep water model for surface gravity waves. (English) Zbl 1534.35313
Summary: We discuss an approximate model for the propagation of deep irrotational water waves, specifically the model obtained by keeping only quadratic nonlinearities in the water waves system under the Zakharov/Craig-Sulem formulation. We argue that the initial-value problem associated with this system is most likely ill-posed in finite regularity spaces, and that it explains the observation of spurious amplification of high-wavenumber modes in numerical simulations that were reported in the literature. This hypothesis has already been proposed by D. M. Ambrose et al. [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 470, No. 2166, Article ID 20130849, 16 p. (2014; Zbl 1371.35210)] but we identify a different instability mechanism. On the basis of this analysis, we show that the system can be “rectified”. Indeed, by introducing appropriate regularizing operators, we can restore the well-posedness without sacrificing other desirable features such as a canonical Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient numerical integration. This provides a first rigorous justification for the common practice of applying filters in high-order spectral methods for the numerical approximation of surface gravity waves. While our study is restricted to a quadratic model, we believe it can be generalized to any order and paves the way towards the rigorous justification of a robust and efficient strategy to approximate water waves with arbitrary accuracy. Our study is supported by detailed and reproducible numerical simulations.
MSC:
35Q35 | PDEs in connection with fluid mechanics |
76B07 | Free-surface potential flows for incompressible inviscid fluids |
76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
76E30 | Nonlinear effects in hydrodynamic stability |
76M22 | Spectral methods applied to problems in fluid mechanics |
35G25 | Initial value problems for nonlinear higher-order PDEs |
35B65 | Smoothness and regularity of solutions to PDEs |
35R25 | Ill-posed problems for PDEs |
35R35 | Free boundary problems for PDEs |
65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Citations:
Zbl 1371.35210References:
[1] | 10.1111/j.1467-9590.2008.00409.x · Zbl 1191.35211 · doi:10.1111/j.1467-9590.2008.00409.x |
[2] | 10.1098/rspa.2013.0849 · Zbl 1371.35210 · doi:10.1098/rspa.2013.0849 |
[3] | 10.1002/sapm1964431309 · Zbl 0128.44601 · doi:10.1002/sapm1964431309 |
[4] | 10.1137/141000671 · Zbl 1356.68030 · doi:10.1137/141000671 |
[5] | 10.1007/s00526-021-01934-6 · Zbl 1471.46030 · doi:10.1007/s00526-021-01934-6 |
[6] | 10.1007/s42286-019-00005-w · Zbl 1451.76022 · doi:10.1007/s42286-019-00005-w |
[7] | 10.1017/S0022112095002011 · Zbl 0920.76013 · doi:10.1017/S0022112095002011 |
[8] | 10.1061/(asce)0733-9399(1999)125:7(756) · doi:10.1061/(asce)0733-9399(1999)125:7(756) |
[9] | 10.1006/jcph.1993.1164 · Zbl 0778.76072 · doi:10.1006/jcph.1993.1164 |
[10] | 10.1088/0951-7715/5/2/009 · Zbl 0742.76012 · doi:10.1088/0951-7715/5/2/009 |
[11] | 10.1017/s002211208700288x · doi:10.1017/s002211208700288x |
[12] | 10.1016/0375-9601(96)00417-3 · doi:10.1016/0375-9601(96)00417-3 |
[13] | ; Hörmander, Lars, Lectures on nonlinear hyperbolic differential equations. Mathématiques & Applications (Berlin), 26, (1997) · Zbl 0881.35001 |
[14] | 10.1007/s00205-012-0604-6 · Zbl 1278.35194 · doi:10.1007/s00205-012-0604-6 |
[15] | 10.1090/surv/188 · doi:10.1090/surv/188 |
[16] | 10.1063/1.3053183 · Zbl 1183.76294 · doi:10.1063/1.3053183 |
[17] | 10.1103/PhysRevLett.69.609 · Zbl 0968.76516 · doi:10.1103/PhysRevLett.69.609 |
[18] | 10.1007/s10231-020-00966-7 · Zbl 1473.46043 · doi:10.1007/s10231-020-00966-7 |
[19] | ; Nicholls, David P., High-order perturbation of surfaces short course: boundary value problems, Lectures on the theory of water waves. London Math. Soc. Lecture Note Ser., 426, 1, (2016) · Zbl 1360.76058 |
[20] | 10.1002/sapm1972513253 · Zbl 0282.65083 · doi:10.1002/sapm1972513253 |
[21] | 10.1007/978-1-4612-5561-1 · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
[22] | 10.1063/1.4738638 · Zbl 1426.76076 · doi:10.1063/1.4738638 |
[23] | 10.1016/j.coastaleng.2007.11.002 · doi:10.1016/j.coastaleng.2007.11.002 |
[24] | 10.1017/S0022112098001219 · Zbl 0911.76011 · doi:10.1017/S0022112098001219 |
[25] | ; Stokes, G. G., On the theory of oscillatory waves, Trans. Cambridge Philos. Soc., 8, 4, 441, (1847) |
[26] | 10.1137/1.9780898719598 · Zbl 0953.68643 · doi:10.1137/1.9780898719598 |
[27] | 10.1029/jc092ic11p11803 · doi:10.1029/jc092ic11p11803 |
[28] | 10.1090/conm/635/12713 · doi:10.1090/conm/635/12713 |
[29] | 10.1007/bf00913182 · doi:10.1007/bf00913182 |
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.