A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations

Coastal Engineering

Volumn 56, Issue 1, 2009, Pages 32-45

  Shiach, J B ^a   Mingham, C G ^a  

^a MANCHESTER METROPOLITAN UNIVERSITY   (United Kingdom)

Author keywords

Boussinesq equations; Finite difference; Finite volume; Godunov type scheme; Wave propagation

Indexed keywords

FLOW OF WATER;

BOUSSINESQ; BOUSSINESQ EQUATIONS; CELL INTERFACES; COMPUTATIONAL RESOURCES; COMPUTATIONAL SPEEDS; DISCRETISATION; FINITE-DIFFERENCE; FINITE-DIFFERENCE METHODS; FINITE-VOLUME; GODUNOV-TYPE SCHEME; GOVERNING EQUATIONS; NUMERICAL MODELS; NUMERICAL RESULTS; NUMERICAL SCHEMES; RECONSTRUCTION TECHNIQUES; RIEMANN PROBLEMS; SPATIAL DERIVATIVES; STANDARD TESTS; TIME USING; WAVE FLUMES;

WAVE EQUATIONS;

BOUSSINESQ EQUATION; FINITE DIFFERENCE METHOD; FINITE VOLUME METHOD; FLUME EXPERIMENT; NUMERICAL MODEL; WAVE PROPAGATION;

EID: 56949106640     PISSN: 03783839     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.coastaleng.2008.06.006     Document Type: Article

References (32)

1. Beji S., and Battjes J.A. Experimental investigations of wave propagation over a bar. Coastal Engineering 19 1-2 (1993) 151-162
2. Causon D.M., Ingram D.M., Mingham C.G., Yang G., and Pearson R.V. Calculation of shallow water flows using a Cartesian cut cell approach. Advances in Water Resources 23 (2000) 545-562
3. Dingemans M.W. Verification of numerical wave propagation models with laboratory measurements. Report H228 Part 1 (1987), Delft Hydraulics 400 pp
4. Dingemans M.W. Comparison of computations with Boussinesq-like models and laboratory measurements. Technical Report H1684.12 (1994), Delft Hydraulics
5. Erduran K.S., Ilic S., and Kutija V. Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equations. International Journal for Numerical Methods in Fluids 49 (2005) 1213-1232
6. Gobbi M.F., and Kirby J.T. Wave evolution over submerged sills: tests of a high-order Boussinesq model. Coastal Engineering 37 (1999) 57-96
7. Harten A., Lax P., and van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review 25 (1983) 35-61
8. Hasselmann K., Barnett T.P., Bouws E., Carlson H., Cartwright D.E., Enke K., Ewing J.A., Gienapp H., Hasselmann D.E., Kruseman P., Meerburg A., Miller P., Olbers D.J., Richter K., Sell W., and W. H. Measurements of wind-wave growth and swell decay during the JOint North Sea WAve Project (JONSWAP). Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe A 8 12 (1973)
9. Hu K., Mingham C.G., and Causon D.M. A bore-capturing finite volume method for open-channel flows. International Journal for Numerical Methods in Fluids 28 (1998) 1241-1261
10. Hu K., Mingham C.G., and Causon D.M. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coastal Engineering 41 (2000) 433-465
11. Hubbard M.E., and Dodd M. A 2D numerical model of wave run-up and overtopping. Coastal Engineering 47 (2002) 1-26
12. Lynett P.J., and Liu P.L.F. Linear analysis of the multi-layer model. Coastal Engineering 51 (2004) 439-454
13. Lynett P.J., Wu T., and Liu P.L.F. Modeling wave runup with depth-integrated equations. Coastal Engineering 46 (2002) 98-107
14. Madsen P.A., and Sørensen O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coastal Engineering 18 (1992) 183-204
15. Madsen P.A., Murray R., and Sørensen O.R. A new form of Boussinesq equations with improved linear dispersion characteristics. Coastal Engineering 15 (1991) 371-388
16. Mingham C.G., and Causon D.M. High-resolution finite-volume method for shallow water flows. Journal of Hydraulic Engineering 124 (1998) 605-614
17. Nwogu O. An alternative form of the Boussinesq equations for nearshore wave propagation. Journal of Waterway, Port, Coastal, and Ocean Engineering 119 6 (1993) 618-638
18. Pearson, J., Bruce, T., and Allsop, N.W.H. (2000). Violent overtopping of waves at seawalls, Experimental results for test series 3a, Private communication. Technical report, University of Edinburgh and HR Wallingford.
19. Peregrine D.H. Long waves on a beach. Journal of Fluid Mechanics 27 4 (1967) 815-827
20. Shi F., Dalrymple R.A., Kirby J.T., Chen Q., and Kennedy A. A fully nonlinear Boussinesq model in generalized curvilinear coordinates. Coastal Engineering 42 (2001) 337-358
21. Shiach J.B., Mingham C.G., Ingram D.M., and Bruce T. The applicability of the shallow water equations for modelling violent wave overtopping. Coastal Engineering 51 (2004) 1-15
22. Shiach J.B., Mingham C.G., Ingram D.M., Causon D.M., Bruce T., Pearson J., and Allsop N.W.H. Extended shallow water prediction of overtopping. In: Smith J.M. (Ed). 29th International Conference on Coastal Engineering, Lisbon, Portugal (2005), National Civil Engineering Laboratory
23. Toro E.F. Riemann problems and the WAF method for solving two-dimensional shallow water equations. Philosophical Transactions of the Royal Society of London, Series A 338 (1992) 843
24. van Leer B. Towards the ultimate conservative difference scheme, V. A second order sequal to Godunov's method. Journal of Computational Physics 32 (1979)
25. van Leer B. On the relation between the upwind-differencing schemes of Godunov, Enguist-Osher and Roe. SIAM Journal of Scientific and Statistical Computing 5 (1985) 1
26. Wei G., and Kirby J.T. Time-dependent numerical code for extended Boussinesq equations. Journal of Waterway, Port, Coastal, and Ocean Engineering 121 5 (1995) 251-261
27. Wei G., Kirby J.T., Grilli S.T., and Subramanya R. A fully nonlinear Boussinesq model for surface waves: part 1. highly nonlinear unsteady waves. Journal of Fluid Mechanics 294 (1995) 71-92
28. Wei G., Kirby J.T., and Sinha A. Generation of waves in Boussinesq models using a source function method. Coastal Engineering 36 (1999) 271-299
29. Witting J.M. A unified model for the evolution of non-linear water waves. Journal of Computational Physics 56 (1984) 203-236
30. Yamamoto S., Kano S., and Daiguji H. An efficient CFD approach for simulating unsteady hypersonic shock-shock interference flows. Computers and Fluids 27 5-6 (1998) 571-580
31. Zhao D.H., Shen H.W., Tabios G.Q., and Lai J.S. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modelling. Journal of Hydraulic Engineering 122 12 (1996) 692-702
32. Zhou J.G., Causon D.M., Mingham C.G., and Ingram D.M. The surface gradient method for the treatment of source terms in the shallow water equations. Journal of Computational Physics 168 (2001) 1-25