# Application of adaptive time marching scheme in exponential basis function method in simulating solitary wave propagation

- 34 Downloads

## Abstract

Solitary wave theories are frequently adopted to simulate the characteristics of tsunami and to predict the resulting run-up in a numerical and experimental manner. A mesh-free method based on exponential basis functions (EBFs) is implemented to simulate the generation and propagation of solitary waves generated by piston-type wave maker. The pressure form of the Euler equations in a Lagrangian formulation is considered in the method. One of the most significant issues in the numerical simulation of solitary wave is to stabilize the wave height during the propagation. A new stable Lagrangian time marching algorithm is introduced which applies an adaptive parameter based on pressure formulation for tracking free surface and to minimize the discrepancies. In order to evaluate the accuracy of this newly introduced scheme, the results of EBF numerical model is applied to wave propagation according to six solitary wave theories. The results indicate the advantage of the introduced adaptive Lagrangian time marching algorithm and the EBF method in simulating solitary wave propagation with pronounced accuracy, convenient performances and the least run time calculation with implications for predicting solitary wave run-up on the beach.

## Keywords

Solitary wave Mesh-free method Pressure formulation Exponential basis functions Adaptive Lagrangian time marching## Notes

## Supplementary material

## References

- 1.Li Y (2000) Tsunamis: non-breaking and breaking solitary wave run-up. PhD thesis, California Institute of TechnologyGoogle Scholar
- 2.Wazwaz AM (2009) Partial differential equations and solitary wave theory. Higher Education Press Springer, BerlinCrossRefGoogle Scholar
- 3.Russel JS (1845) Report on waves. In: Proceeding of 14th meeting, British association for the advancement of science, pp 311–390Google Scholar
- 4.Daily JW, Stephan SC (1952) The solitary wave: its celerity, internal velocity and amplitude attenuation in a horizontal smooth channel. In: Proceeding of 3rd conference, coastal engineering, pp 13–30Google Scholar
- 5.Wiegel RL (1955) Laboratory studies of gravity waves generated by the movement of a submarine body. EOS Trans Am Geophys Union 36:759–774CrossRefGoogle Scholar
- 6.Kishi T, Saeki H (1966) The shoaling, breaking, and runup of the solitary wave on impermeable rough slopes. In: Proceeding of 10th conference, coastal engineering, pp 322–348Google Scholar
- 7.Camfield FE, Street RL (1969) Shoaling of solitary waves on small slopes. In: Proceedings of ASCE, WW95, pp 1–22Google Scholar
- 8.Goring DG (1979) Tsunami: the propagation of long waves onto a shelf. PhD thesis, California Institute of TechnologyGoogle Scholar
- 9.Renouard DP, Seabra FJ, Temperville AM (1985) Experimental study of the generation, damping, and refelxion of a solitary wave. Dyn Atmos Oceans 9:341–358CrossRefGoogle Scholar
- 10.Guizien K, Barthelemy E (2002) Short waves modulations by large free surface solitary waves. Experiments and models. Phys Fluids 13:3624–3635CrossRefGoogle Scholar
- 11.Boussinesq MJ (1871) Théorie de l’intumescence liquide, appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. C R Acad Sci Paris 72:755–759Google Scholar
- 12.Grilli S, Svendsen IA (1991) The propagation and runup of solitary waves on steep slopes. Report No. 91-4, Center for Applied Coastal Research, University of Delaware, Newark, DelawareGoogle Scholar
- 13.Liu PL-F, Park YS, Cowen EA (2007) Boundary layer flow and bed shear stress under a solitary wave. J Fluid Mech 574:449–463CrossRefGoogle Scholar
- 14.Rayleigh L (1876) On waves. Philos Mag 1:257–279CrossRefGoogle Scholar
- 15.Clamond D, Germain J-P (1999) Interaction between a Stokes wave packet and a solitary wave. Eur J Mech B/Fluids 18(1):67–91CrossRefGoogle Scholar
- 16.Temperville A (1985) Contribution à l’étude des ondes de gravité en eau peu profonde. Thèse d’Etat, Université Joseph Fourier—Grenoble IGoogle Scholar
- 17.Malek-mohammadi S, Testik FY (2010) New methodology for laboratory generation of solitary wave. J Waterw Port Coast Ocean Eng ASCE 136:286–294CrossRefGoogle Scholar
- 18.Wu NJ, Tsay TK, Chen Y (2014) Generation of stable solitary waves by a piston-type wave maker. Wave Motion 51:240–255CrossRefGoogle Scholar
- 19.Grimshaw RHJ (1971) The solitary wave in water of variable depth, part 2. J Fluid Mech 46:611–622CrossRefGoogle Scholar
- 20.Fenton J (1972) A ninth-order solution for the solitary wave. J Fluid Mech 52:257–271CrossRefGoogle Scholar
- 21.Zandi SM, Rafizadeh A, Shanehsazzadeh A (2017) Numerical simulation of non-breaking solitary wave run-up using exponential basis functions. Environ Fluid Mech 7(5):1015–1034CrossRefGoogle Scholar
- 22.Boroomand B, Bazazzadeh S, Zandi SM (2016) On the use of Laplace’s equation for pressure and a mesh-free method for 3D simulation of nonlinear sloshing in tanks. Ocean Eng 122:54–67CrossRefGoogle Scholar
- 23.Zandi SM, Boroomand B, Soghrati S (2012) Exponential basis functions in solution of incompressible fluid problems with moving free surfaces. J Comput Phys 231:505–527CrossRefGoogle Scholar
- 24.Zandi SM, Boroomand B, Soghrati S (2012) Exponential basis functions in problems with fully incompressible materials: a mesh-free method. J Comput Phys 231:7255–7273CrossRefGoogle Scholar
- 25.Zandi SM (2014) Non-linear analysis of free surface problems with moving boundaries by meshless local exponential basis functions method. PhD thesis, Isfahan University of TechnologyGoogle Scholar