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A Novel Finite Element Scheme of Nwogu Extended Boussinesq Equations to Predict Free Surface Elevation over Different Bathymetry of Beaches

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In this study, free surface elevation is predicted by using a new finite element scheme. This numerical method solves a Nwogu Boussinesq equation system to simulate wave propagation in the complicated bathymetry of coastal regions. The numerical
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   A Novel Finite Element Scheme of Nwogu ExtendedBoussinesq Equations to Predict Free Surface Elevationover Different Bathymetry of Beaches  Arsham Reisinezhad † , Said Mazaheri † * , Kourosh Hejazi ‡ , and Mohammad Hadi Jabbari ‡ † Ocean Engineering and Technology Research CenterIranian National Institute for Oceanography and Atmospheric ScienceTehran Iran ‡ Civil Engineering FacultyK.N. Toosi University of TechnologyTehran Iran  ABSTRACT Reisinezhad, A.; Mazaheri, S.; Hejazi, K., and Jabbari, M.H., 0000. A novel finite element scheme of Nwogu extendedBoussinesq equations to predict free surface elevation over different bathymetry of beaches.  Journal of Coastal Research, 00(0), 000–000. Coconut Creek (Florida), ISSN 0749-0208.In this study, free surface elevation is predicted by using a new finite element scheme. This numerical method solves aNwogu Boussinesq equation system to simulate wave propagation in the complicated bathymetry of coastal regions. Thenumerical approach is based on a Galerkin finite element approach for spatial discretization and Adam-Bashforth-Moulton predictor-corrector strategy for time integration. Governing equations are rewritten in lower-order forms byintroducing a novel form of auxiliary variable in order to make the application of the linear finite element methodpossible. Then, the stability of the suggested finite element schemes is studied using a theoretical analysis. For thevalidation of the present numerical method, five test cases are considered to show the capability of the numerical modelfor simulating the free surface elevation of wave propagation over different beach profiles where the nonlinear anddispersive effects are so important. The simulated results agree well with experimental observations.  ADDITIONAL INDEX WORDS:  Coastal hydrodynamics, Adam-Bashforth-Moulton predictor-corrector scheme, slopingand barred beaches . INTRODUCTION Water surface elevation around coastal structures is one of the important parameters requiredtodesign protectivecoastalstructures. The form of bathymetry as well as the geometry of protective structures (  e.g.,  breakwaters) are counted as mainfactorsthat influence the water surface elevationsnear coastalstructures. Moreover, in nearshore regions where nonlinearanddispersivecharacteristicsofwavesarestronglysignificant,the variation of water surface elevations in the presence of waves becomes very important in ocean engineering problems. As an alternative to costly physical experiments, numericalmodels are cheaper and more flexible tools that can be used tosimulate the hydrodynamic phenomena as well as watersurfacevariationsaroundcoastalstructures.Inrecentdecades,several mathematical equations have been developed andproposed to present wave hydrodynamic processes in near-shore zones. Because of the computational constraints, such astime-domain and large-scale wave evolution, Boussinesq-typeequationscanbeusedasoneofthemostappropriatemodelsforthis task.The Boussinesq-type models have been derived based on thedepth-averaged velocity. The first standard Boussinesq model,derived by Peregrine (1967), has good linear accuracy only upto  kh ’ 0.75 (where  k  and  h  are the wave number and waterdepth, respectively). In subsequent decades, improved Boussi-nesq equations were presented by Madsen, Murray, andSørensen (1991), Nwogu (1993), and Beji and Nadaoka(1996). These new forms of Boussinesq equations, which arecalled extended Boussinesq equations, can be applied not onlyfornearshoreregions,butalsoforintermediatewaterdepth,  kh   3, where the majority of protective coastal and nearshorestructures exist.From the mathematical viewpoint, the Nwogu Boussinesqequation system is derived from the continuity equation andEuler’s equations of motion (Ghadimi, Jabbari, and Reisinez-had, 2012) by considering the third-order velocity vectorderivatives in the continuity equation. Due to the relativelystraightforward form of the Nwogu Boussinesq equationsystem, this system of equations has been used more thanother systems of equations by researchers in the relevant fieldof work. Accordingly, different numerical methods have beenapplied to solve the Nwogu extended Boussinesq equations(NBE). Nwogu (1993)discretized the NBE by the use of a finitedifference scheme and added a corrector term to eliminatenonphysical dispersion. He used this numerical scheme tomodel regular and irregular wave shoaling on the beach andshowedthat his equations couldbeused toaccuratelysimulateregular and irregular wave trains. Lin and Man (2007)presented a staggered grid finite difference method with afourth-order predictor-corrector scheme for time integrationand applied it to model the water sloshing in a container, wavepropagationovera bar ina one-dimensionaldomain,and otherstandard tests in two-dimensional domains. A finite elementscheme based on the Galerkin method for solution of theseequations was first developed by Walkley and Berzins (1999).They rewrote NBE in lower-order form by introducing an  DOI: 10.2112/JCOASTRES-D-14-00041.1 received 3 March 2014;accepted in revision 22 October 2014; corrected proofs received 22 December 2014; published pre-print online XX Month XXXX. *Corresponding Author: said.mazaheri@inio.ac.ir  Coastal Education and Research Foundation, Inc. 2015  //titan/production/c/coas/live_jobs/coas-32-01/coas-32-01-10/layouts/coas-32-01-10.3d  9 January 2015  8:00 am  Allen Press, Inc.  Customer MS# JCOASTRES-D-14-00041.1 Page 1 Journal of Coastal Research 00 0 000–000 Coconut Creek, Florida Month 0000  auxiliary equation. Meanwhile, for time integration of dis-cretized terms, they used the Sprint software package. Theysimulated solitary and periodic wave propagation over asloping beach and over a bar, respectively. Woo and Liu(2004) also employed the Galerkin finite element scheme andreduced the order of Nwogu Boussinesq equations using anauxiliary equation and performed some standard tests. Codina  et al . (2008) also presented a new finite element method. Theirformulation was based on the variation of a multiscaleapproach. The basic idea was to split the unknowns into aresolvable component, which can be reproduced by thediscretization method, and a remainder, which they calledsubgrid scale or subscale. They applied their numerical code toa realistic practical work.The aim of this paper is to propose an efficient numericalmodel for prediction of the water surface elevation of wavepropagation affected by barred beaches in intermediate waterdepths of coastal regions. The model has been developed tosimulate the nonlinearity and dispersion properties of wavepropagation on a complicated gentle slope bathymetry locatedbeforethebreakingzone.Accordingly,anewnumericalschemebased on the finite element formulation, as well as the Adams-Bashforth-Moulton predictor-corrector method (Ghadimi, Jab-bari, and Reisinezhad, 2011), is presented to discretize a one-dimensional spatial domain and temporal domain, respective-ly.Thediscretizationofbothspatialandtemporaldomainsandthe introduction of a new auxiliary variable to simplify thehigh-order terms of the Nwogu Boussinesq equations are themain theoretical novelties of this model in comparison withother available models mentioned earlier, such as that of Walkley and Berzins (1999).In order to verify the proposed numerical method, evaluatethe capability of the model, and analyze the accuracy of simulated water surface elevations due to wave propagation,five different cases are considered in coastal regions. In thesetest cases, dispersive and nonlinearity effects of the periodicwaves over sloping and barred beaches are investigated. Thiscan be regarded as an additional specific applicability of themodel in respect to other models such as that of Walkley andBerzins (1999). Outputs are compared with experimentalobservations and results of other existing numerical models. METHODS In this section, first the governing equations as well as thefinite element formulation of the proposed problem arepresented and discussed in detail, and then the stability of the numerical model is analyzed. Governing Equation The physical system that depicts the free surface and bottomseabed can be shown by  H  ,  k , and  a , which are typical waterdepth, typical wave length, and typical wave amplitude,respectively. The nonlinearity    and dispersion  r  present inthe system are parameterized as follows:   ¼  a H   ð 1 Þ r  ¼  H  k  ð 2 Þ  Assumingthat  ,, 1 and  r ,, 1 (  e.g., longperiodicwaves),the NBE system of equations is derived by integration throughthe depth of the irrotational, incompressible, and inviscid fluidflow equations. Accordingly, the one-dimensional Nwoguextended Boussinesq equations are presented as follows interms of free surface  g (  x ,  y ), velocity  u (  x ,  t ) at depth  z a , andprofile  h (  x ): g t  þ  ]]  x  ð h þ g Þ u ½ þ  ]]  x z 2 a 2   h 2 6   h  ] 2 u ]  x 2   þ  z a  þ h 2   h  ] 2 ð hu Þ ]  x 2    ¼  0  ð 3 Þ u t  þ u ] u ]  x  þ  g ] g ]  x þ  z 2 a 2 ] 2 u t ]  x 2  þ  z a ] 2 ]  x 2  ð hu t Þ ¼  0  ð 4 Þ where  z a  ¼  h h  ð 5 Þ in which  z a  is a free parameter.The value of   h  is taken to be  h ¼ 0.531 throughout this work, as suggested by Nwogu (1993).The values of free parameters in each extended Boussinesqequation system are obtained from the best agreementachievedbetweenthelinearizedandexactdispersionrelations. After introducing a new Boussinesq equation system, theequations are linearized, and phase celerity ( C ) or groupvelocities are compared to the exact dispersion relation orStokes first-order theory ( C 2 ¼  gh [tanh  kh ]/[  kh ]).The linearized dispersion relation property of the NBEaccording to Nwogu (1993) is given as: C 2 ¼  gh  1  ð a  þ  1 = 3 Þð  kh Þ 2 1   a ð  kh Þ 2 " #  ð 6 Þ InEquation(6),  a ¼ 0.390, which corresponds tothe velocityat an elevation of   z a ¼ 0.531 h  (Nwogu, 1993). Numerical Formulation  Application of a linear finite element method from theviewpoint of simplicity in formulation is similar to theapplication of a central finite difference method, but with thisdifference: The finite element can also be applied to theunstructured mesh. Due to the presence of third-order spatialderivatives in Equation (3), the linear Galerkin finite elementmethod cannot be applied directly to the equation system. A weighted residual form of the problem will contain second-order spatial derivatives (second-order velocity derivatives inNBE), in which case the linear finite element approximationcannot be applied. Therefore, special treatment of the second-order derivatives becomes necessary. A model based on the linear Galerkin finite element methodfor NBE was first offered by Walkley and Berzins (1999). Toovercome the difficulty associated with the third-order spatialderivatives in NBE, they introduced an auxiliary variable andrewrote the equation system in its lower-order form. Applyingthis auxiliary equation, the high-order terms of the NwoguBoussinesq equations are eliminated, and thus the solution of NBE by linear elements becomes possible.Using the mathematical form of Equation (3), the auxiliaryequation introduced by Walkley and Berzins (1999) has the  //titan/production/c/coas/live_jobs/coas-32-01/coas-32-01-10/layouts/coas-32-01-10.3d  9 January 2015  8:00 am  Allen Press, Inc.  Customer MS# JCOASTRES-D-14-00041.1 Page 2 Journal of Coastal Research, Vol. 00, No. 0, 00000 Reisinezhad  et al.  form w   z 2 a 2   h 2 6   h  ] 2 u ]  x 2   þ  z a  þ h 2   h  ] 2 ð hu Þ ]  x 2     ¼  0  ð 7 Þ DespitethemodelproposedbyWalkleyandBerzins(1999),inthis study, a flexible method for choosing the auxiliary variableis introduced that is not restricted by the mathematical form of thegoverningequations.Inordertointroducethenewauxiliaryvariable, it is necessary that the third term on the left side of Equation (3) be spatially differentiated so that the auxiliaryvariable, which is naturally derived from this equation, isomitted. In the finite element scheme, by defining a set of   N  nodes as the intersection nodes of (  N   – 1) nonoverlappingelements, the spatial domain  X  is covered completely. Nowassumethatthevaluesofdepthandseabedslopeofeachrelatedelementareconstant.Therefore,withthisknowledgeaswellasEquation (5), Equation (3) is written as follows: g t  þ  ]]  x  ð h þ g Þ u ½  þ  ]]  x  ð  A 1  þ  A 2 Þ h 3 ] 2 u ]  x 2  þ ð 3  A 1  þ  5  A 2 Þ h 2 ] h ]  x ] u ]  x  þ 4  A 2 h  ] h ]  x   2 u g  ¼  0  ð 8 Þ where  A 1  and  A 2  are given as  A 1  ¼  h 2 2    16 ;  A 2  ¼  h  þ  12  ð 9 Þ Despite the auxiliary variable presented by Walkley andBerzins (1999), the new auxiliary variable introduced in thisnumerical method is assumed equal to the last term of Equation (8): w  ð  A 1  þ  A 2 Þ h 3 ] 2 u ]  x 2  þ ð 3  A 1  þ  5  A 2 Þ h 2 ] h ]  x ] u ]  x  þ 4  A 2 h  ] h ]  x   2 u ( ) ¼  0 ð 10 Þ It is quite clear that Equation (7) and Equation (10) havedifferent mathematical forms. Because, in this paper, to obtaina new extra equation for NBE, the partial differential terms of NBE in the continuous domain are reformulated by using themathematical properties of the discrete domain. This meansthat, before discretizing the governing equations by using afinite element method, the continuity equation of the NBE wasmanipulated in the sense that the value of water depth isassumed constant at each node of the spatial discretizeddomain. Consequently, spatial derivative terms of NBE can berewritteninanother mathematicalform.Forinstance, ] ( hu )/  ]  x willbeequalto h ] ( u )/  ]  x ,eveniftheseabedhasavariabledepth. Anovelformoftheextraequationisintroducedbyapplyingthedescribed assumption. Consequently, the presented equationlets us apply a linear finite element scheme for NBE andsimplifies the solution to the governing equations.Now, by considering Equation (10) as an auxiliary equation,the system of equations is reduced to its lower-order form: g t  þ  ]]  x  ð h þ g Þ u ½  þ  ]]  x ð w Þ ¼  0  ð 11 Þ u t  þ u ] u ]  x  þ  g ] g ]  x þ  B 1 h 2 ] 2 u t ]  x 2  þ  B 2 h  ] 2 ]  x 2  ð hu t Þ ¼  0  ð 12 Þ w  ð  A 1  þ  A 2 Þ h 3 ] 2 u ]  x 2  þ ð 3  A 1  þ 5  A 2 Þ h 2 ] h ]  x ] u ]  x  þ  4  A 2 h  ] h ]  x   2 u ( ) ¼  0 ð 13 Þ where  B 1  and  B 2  are as follows:  B 1  ¼  h 2 2  ;  B 2  ¼  h  ð 14 Þ The dependent variables in Equations (11–13) are approxi-mated within the finite elements as follows: v ’ X  N i ¼ 1  N  i v i  ð 15 Þ where  v i  are the values of any dependent variable ( u ,  g , and  w )at the nodal points, and  N  i  is the standard basis function. Thelinear Galerkin finite element method is developed bymultiplying Equations (11–13) in the trial function  N  i (  x ) takenfrom the set of linear basis functions and integrating over eachelement of the spatial domain  X . Also, integration by parts isused to reduce the second-order spatial derivatives (see Appendix). The resulting element matrix equation for anelement can also be obtained as follows:  M   ki ˙ g i  þ Q  kij  ð h þ g Þ i u  j   þ Q  kij  u i ð h þ  g Þ  j   þ T   ki w i  ¼  0  ð 16 Þ  M   ki  ˙u i  þ Q  kij  u i u  j   þ  gT   ki g i   ð  B 1  þ  B 2 Þ h 2  E  ki  ˙u i  þ 2  B 2 hh  x T   ki  ˙u i þ ð  B 1  þ  B 2 Þ h 2  N   k  ˙u  x ½   x i þ 1  x i ¼  0  ð 17 Þ  M   ki w i   ð  A 1  þ  A 2 Þ h 3  E  ki u i  þ ð 3  A 1  þ 5  A 2 Þ h 2 h  x T   ki u i þ 4  A 2 hh 2  x u i  þ ð  A 1  þ  A 2 Þ h 3  N   k u  x ½   x i þ 1  x i ¼  0  ð 18 Þ where  M   ki  ¼ Z  X  N   k  N  i dx  ð 19 Þ Q  kij   ¼ Z  X  N   k  N  i dN   j  ]  x dx  ð 20 Þ T   ki  ¼ Z  X  N   k dN  i dx dx  ð 21 Þ  E  ki  ¼ Z  X dN   k dxdN  i dx dx  ð 22 Þ In Equations 16 and 17, the dot over a variable denotes thepartial differentiation with respect to time, and  h  x  is thederivative of the water depth with respect tospatial coordinate  //titan/production/c/coas/live_jobs/coas-32-01/coas-32-01-10/layouts/coas-32-01-10.3d  9 January 2015  8:00 am  Allen Press, Inc.  Customer MS# JCOASTRES-D-14-00041.1 Page 3 Journal of Coastal Research, Vol. 00, No. 0, 0000Novel Nwogu Extended Boussinesq Equations for Beaches 0   x .TheintegralsinEquations(19)through(22)areevaluatedbyintegrating over each element individually.The boundary terms appearing in these equations vanishidentically, except for  i ¼ 1 and  i ¼  N  , where  i  is the number of each node. If Dirichlet boundary conditions are specified at theboundary, the corresponding boundary terms are eliminated(Katopodes and Wu, 1986).Finally, after discretization, the assembled global forms of Equations (16) and (17) take the form:  M  ½   ˙ f     ¼  E 1 f g ð 23 Þ  Also, the assembled global form of the auxiliary equation(Equation [18]) is obtained as follows:  M  ½   w f g ¼  E 2 f g ð 24 Þ In Equations (23) and (24),  M   is the coefficient matrix,  f   isequal to the free surface elevation  g  or velocity  u , while  E 1  and  E 2  are vectorsthat are determinedby the known values of  u ,  g ,and w .TheSYSTEMoflinearequationsinEquation(24)canbesolved explicitly at each time step, and Equation (23) must beintegrated in time by a typical high-order method. Time Scheme Description If the variable  f   is discretized at  t  ¼  n D t , the Adams-Bashforth-Moulton (ABM) predictor-corrector scheme is usedfor time integration in the following form (Bellotti andBrocchini, 2001):1. Evaluation of the right-hand side of Equation (23) at timelevels  n ,  n  – 1, and  n  – 22. Integration in time of Equation (23) by means of thepredictor stage of the ABM scheme  M  ½   f  f g n þ 1 ¼  M  ½   f  f g n þ D t 12 23  E 1 f g n  16  E 1 f g n  1 þ 5  E 1 f g n  2 h i ð 25 Þ 3. Evaluation of the right-hand side of Equation (23) at timelevel  n  þ  1.4. Integration in time of Equation (23) by means of thecorrector stage of the ABM scheme:  M  ½   f  f g n þ 1 ¼  M  ½   f  f g n þ D t 24 9  E 1 f g n þ 1 þ 19  E 1 f g n  5  E 1 f g n  1 þ  E 1 f g n  2 h i ð 26 Þ Steps 3–4 are iterated until convergence is reached. Boundary Conditions Inflow and outflow boundaries are two types of generalboundaries that are employed here. Inflow Boundary Condition In this case, the initial free surface elevation is set to be zeroall over the domain, and the elevation  g  at the boundary isprescribed at each time step as follows: g ð  x ; t Þ ¼  a sin ð  kx  x t Þ ð 27 Þ For the incident wave elevation, the linear theory (Dean andDalrymple,1991)isusedtoobtaintheincidentwavevelocityasin u  ¼  x  kh g  ð 28 Þ The system of discretized equations becomes a complete setbyadditionofaspecificvariableof  w ontheinflowboundaryforeach model. Therefore,  w i  for the NBE equation is determinedas follows: w i   a x  kh  ð  A 1  þ  A 2 Þ  k 2 h 3 sin ð  kx  x t Þ ½ þð 3  A 1  þ  5  A 2 Þ  kh 2 ] h ]  x cos ð  kx   x t Þþ 4  A 2 h  ] h ]  x   2 sin ð  kx   x t Þ  ¼  0  ð 29 Þ Outflow Boundary Condition Minimization of the nonphysical reflection of informationback into the domain is important at the outflow or shorelineboundary.Therefore,inthisstudy,aviscousdampinglayer, i.e .a spongelayer, proposedbyLarsenand Dancy (1983),isplacedin front of the outflow or shoreline boundary to absorb theincoming wave energy. On the sponge layer, the surfaceelevation  g  and the velocity  u  are divided by  l (  x ) after eachtime step. The factor  l (  x ) takes the following form: l ð  x Þ ¼  exp  ð 2  d = D d   2  d s = D d Þ ln ð a Þ h i 0    d    d s 1  d , d s (  ð 30 Þ where  d  is the distance between the boundary and the point onthespongelayer, D d isthetypicaldimensionoftheelements, d s is usually equal to one or two wave lengths, and  a  is a constanttobespecified.Inthepresentwork,accordingtothe work doneby Li  et al . (1999), a value of   a ¼ 4 is proposed. Stability Analysis In this part, the stability of the numerical model is analyzedby the classical Von-Neumann analysis (Lin and Man, 2007).It is important to underline that the Von Neumann stabilityanalysis is local in the sense that: (1) it does not take intoaccount boundary effects; and (2) it assumes that thecoefficients of the finite element equations, which are suffi-ciently slowly varying, can be considered constant in time andspace.For the stability analysis of NBE, a wave over a constantwater depth is considered with small amplitude (linear). Thus,Equations (11–13) can be reduced to: ˙ g  þ h ] u ]  x  þ ] w ]  x  ¼  0  ð 31 Þ ˙u þ  g ] g ]  x  ð  B 1  þ  B 2 Þ h 2 ] 2 ˙u ]  x 2  ¼  0  ð 32 Þ w  ð  A 1  þ  A 2 Þ h 3 ] 2 u ]  x 2  ¼  0  ð 33 Þ The element matrix equations are presented as follows:  M  ij  ˙ g  j   ¼  hT  ij  u  j    T  ij  w  j   ð 34 Þ  //titan/production/c/coas/live_jobs/coas-32-01/coas-32-01-10/layouts/coas-32-01-10.3d  9 January 2015  8:00 am  Allen Press, Inc.  Customer MS# JCOASTRES-D-14-00041.1 Page 4 Journal of Coastal Research, Vol. 00, No. 0, 00000 Reisinezhad  et al.   M  ij    ð  B 1  þ  B 2 Þ h 2  E ij    ˙u  j   ¼   gT  ij  g  j   ð 35 Þ  M  ij  w  j   ¼ ð  A 1  þ  A 2 Þ h 3  E ij  u  j   ð 36 Þ Under these assumptions, the solution can be seen as a sumof eigenmodes, which at each grid point have the form g ni  ¼  g 0 G n  e  I  h i ð 37 Þ u ni  ¼  u 0 G n  e  I  h i ð 38 Þ w ni  ¼  w 0 G n  e  I  h i ð 39 Þ where  I   ¼  ffiffiffiffiffiffiffi  1 p   is the imaginary unit,  h  ¼  k D  x  is the phaseangle,  k is the spatial wave number, ( g 0 , u 0 ) istheeigenvectorof the problem, the complex number of   G , which is the VanNeumann stability criterion, is named the amplificationfactor, and  n  is time level ( i.e . time scheme description). Bysubstituting Equation (34) into the Adams-Bashforth predic-tor method (Equation [25]), we can obtain the followingequation. D  x 6  g n þ 1 i  1  þ  4 g n þ 1 i  þ  g n þ 1 i þ 1    ¼  D  x 6  g ni  1  þ 4 g ni  þ g ni þ 1     D t 24 23  h u ni þ 1   u ni  1    þ  w ni þ 1   w ni  1     16  h u n  1 i þ 1   u n  1 i  1   þ  w n  1 i þ 1   w n  1 i  1    þ 5  h u n  2 i þ 1   u n  2 i  1   þ  w n  2 i þ 1   w n  2 i  1   g ð 40 Þ Introducing Equations (37) and (38) into Equation (40) willresult in b 1 G 2 ð G   1 Þ g 0  þ b 2 ð 23 G 2   16 G þ  5 Þð hu 0  þ w 0 Þ ¼  0  ð 41 Þ where  b 1  and  b 2  are given as in b 1  ¼  2cos h þ  4  ð 42 Þ b 2  ¼  D t 4 D  x ð 2sin h Þ  I   ð 43 Þ On the other hand, Equation (36) will be expanded at everytime step as follows: D  x 6  ð w ni  1  þ 4 w ni  þ w ni þ 1 Þ  ð  A 1  þ  A 2 Þ h 2 D  x ð u ni  1   2 u ni  þ u ni þ 1 Þ ¼  0 ð 44 Þ ByapplyingEquations(37) and (39), itcan beconcludedthat w 0  ¼  b 3 u 0  ð 45 Þ where b 3  ¼  6 ð  A 1  þ  A 2 Þ  h 3 ð D  x Þ 2 2cos h    22cos h  þ  4  ð 46 Þ By substituting Equation (45) into Equation (41), we canrewrite Equation (41) in the following form: b 1 G 2 ð G  1 Þ g 0  þ b 4 ð 23 G 2  16 G þ  5 Þ u 0  ¼  0  ð 47 Þ where b 4  ¼  b 2 ð h þ b 3 Þ ð 48 Þ  Also,similartotheaboveschemeforEquation(34),Equation(35) becomes b 5 G 2 ð G   1 Þ u 0  þ  b 6 ð 23 G 2  16 G þ 5 Þ g 0  ¼  0  ð 49 Þ where b 5  ¼ ð 2cos h  þ 4 Þ þ  6 ð  B 1  þ  B 2 Þ h 2 ð D  x Þ 2  ð 2cos h   2 Þ ð 50 Þ b 6  ¼  g D t 4 D  x ð 2sin h Þ  I   ð 51 Þ To have nontrivial solutions for eigenvector  ð g 0 ;  u 0 Þ  forEquations (46) and (49), we must have: G 2 ð G   1 Þ   2  b 2 b 6 b 1 b 3 23 G 2  16 G þ 5   2 ¼  0  ð 52 Þ The above equation can be solved numerically for  j G j  as afunction of  h , bysetting D  x ¼ 0.1 h and Cr ¼ (0.5,1.0, 1.5),where Cr  is the Courant number,  i.e . Cr  ¼  ffiffiffiffiffiffi  gh p   D t D  x    ð 53 Þ The Courant number is used to determine the time step  D t once  ffiffiffiffiffiffi  gh p   is known and  D  x  has been chosen to achieve certainaccuracy. So the requirement for the amplification factor,which is the Van Neumann stability criterion,  j G j 2   1, issatisfied as long as  j Cr j   1. The largest modulus of theamplification factor as a function of   h , which determines theCourant number limitation, is shown in Figure 1. It is obviousthat the predictor scheme is stable when the Courant numberis smaller than or equal to unity.The stability of the Adams-Moulton corrector method(Equation [26]) is investigated by following the same proce-dure. The stability properties of the corrector scheme aredetermined by the following equation: G 2 ð G   1 Þ   2  a 4 a 6 a 1 a 5 9 G 3 þ 19 G 2  5 G þ  1   2 ¼  0  ð 54 Þ where a 1  ¼  2cos h  þ 4  ð 55 Þ a 2  ¼  D t 8 D  x ð 2sin h Þ  I   ð 56 Þ a 3  ¼  b 3  ð 57 Þ a 4  ¼  a 4 ð h þ  a 2 Þ ð 58 Þ a 5  ¼ ð 2cos h  þ  4 Þ þ  6 ð  B 1  þ  B 2 Þ h 2 ð D  x Þ 2  ð 2cos h   2 Þ ð 59 Þ a 6  ¼  g D t 8 D  x ð 2sin h Þ  I   ð 60 Þ  //titan/production/c/coas/live_jobs/coas-32-01/coas-32-01-10/layouts/coas-32-01-10.3d  9 January 2015  8:00 am  Allen Press, Inc.  Customer MS# JCOASTRES-D-14-00041.1 Page 5 Journal of Coastal Research, Vol. 00, No. 0, 0000Novel Nwogu Extended Boussinesq Equations for Beaches 0
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