Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite-volume and finite-difference methods is introduced for the solution of the Boussinesq equations. While the finite-volume method with a Riemann solver is applied to the conservative part of the equations, the higher-order Boussinesq terms are discretized using the finite-difference scheme. Fourth-order accuracy in space for the finite-volume solution is achieved using the MUSCL-TVD scheme. Within this, four limiters have been tested, of which van-Leer limiter is found to be the most suitable. The Adams-Basforth third-order predictor and Adams-Moulton fourth-order corrector methods are used to obtain fourth-order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model HYWAVE, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi-chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  Derek M. Causon,et al.  HIGH-RESOLUTION FINITE-VOLUME METHOD FOR SHALLOW WATER FLOWS , 1998 .

[2]  K. S. Erduran,et al.  Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes , 2002 .

[3]  Satoru Yamamoto,et al.  An efficient CFD approach for simulating unsteady hypersonic shock-shock interference flows , 1998 .

[4]  Martin Berzins,et al.  A FINITE ELEMENT METHOD FOR THE ONE-DIMENSIONAL EXTENDED BOUSSINESQ EQUATIONS , 1999 .

[5]  M. Vázquez-Cendón.  Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

[6]  D. Zhao,et al.  Finite‐Volume Two‐Dimensional Unsteady‐Flow Model for River Basins , 1994 .

[7]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[8]  Seung-Buhm Woo,et al.  A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation , 2001 .

[9]  D. Zhao,et al.  Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling , 1996 .

[10]  Derek M. Causon,et al.  Calculation of unsteady bore diffraction using a high resolution finite volume method , 2000 .

[11]  Sandra Soares Frazao,et al.  Undular bores and secondary waves -Experiments and hybrid finite-volume modelling , 2002 .

[12]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[13]  Michael B. Abbott,et al.  Accuracy of Short-Wave Numerical Models , 1984 .

[14]  Yan Yu,et al.  Numerical modeling of Boussinesq equations by finite element method , 1999 .

[15]  Derek M. Causon,et al.  Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations , 2000 .

[16]  M. W. Dingemans,et al.  Water Wave Propagation Over Uneven Bottoms , 1997 .

[17]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry , 1992 .

[18]  Th. V. Karambas,et al.  A breaking wave propagation model based on the Boussinesq equations , 1992 .

[19]  D. Causon,et al.  The surface gradient method for the treatment of source terms in the shallow-water equations , 2001 .

[20]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[21]  Pilar García-Navarro,et al.  Flux difference splitting for 1D open channel flow equations , 1992 .

[22]  C. G. M. A. D. Causon.  On High Resolution Finite Volume Modelling OfDiscontinuous Solutions Of The Shallow WaterEquations , 1997 .

[23]  Jean-Antoine Désidéri,et al.  Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes , 1998 .

[24]  Brett F. Sanders,et al.  Finite-Volume Models for Unidirectional, Nonlinear, Dispersive Waves , 2002 .

[25]  Yan Yu,et al.  Wave concentration by a navigation channel , 2000 .

[26]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics , 1991 .

[27]  Pilar García-Navarro,et al.  A numerical model for the flooding and drying of irregular domains , 2002 .

[28]  Martin Berzins,et al.  A finite element method for the two‐dimensional extended Boussinesq equations , 2002 .

[29]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[30]  K. Nadaoka,et al.  A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth , 1996 .

[31]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[32]  G. Wei,et al.  Time-Dependent Numerical Code for Extended Boussinesq Equations , 1995 .

[33]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .