The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and...The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.展开更多
In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries....In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended <em>F</em>-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.展开更多
A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing e...A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer （HLL） Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth- order monotone upstream-centered scheme for conservation laws （MUSCL）. The time marching scheme based on the third-order TVD Runge- Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.展开更多
In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and ...In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.展开更多
This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are result...This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.展开更多
In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,a...In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid_layer and the reduced kinetic thickness of upper fluid_layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single_layer fluid are extended to the case of stratified fluid.展开更多
It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water eq...It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.展开更多
The complete analytical solution of the Riemann problem for the homo-geneous Dispersive Nonlinear Shallow Water Equations [Antuono, Liapidevskii andBrocchini, Stud. Appl. Math., 122 (2009), pp. 1-28] is presented, for...The complete analytical solution of the Riemann problem for the homo-geneous Dispersive Nonlinear Shallow Water Equations [Antuono, Liapidevskii andBrocchini, Stud. Appl. Math., 122 (2009), pp. 1-28] is presented, for both wet-bed anddry-bed conditions. Moreover, such a set of hyperbolic and dispersive depth-averagedequations shows an interesting resonance phenomenon in the wave pattern of the solu-tion and we define conditions for the occurrence of resonance and present an algorithmto capture it. As an indirect check on the analytical solution we have carried out a de-tailed comparison with the numerical solution of the government equations obtainedfrom a dissipative method that does not make explicit use of the solution of the localRiemann problem.展开更多
文摘The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations （NSWE） for the wave run-up on a beach. The finite volume method （FVM） is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe＇ s flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws （MUSCL） variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.
文摘In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended <em>F</em>-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.
基金supported by the National Natural Science Foundation of China(Grant No.51579034)the Open Fund of the Key Laboratory of Ocean Circulation and Waves,Chinese Academy of Sciences(Grant No.KLOCW1502)
文摘A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer （HLL） Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth- order monotone upstream-centered scheme for conservation laws （MUSCL）. The time marching scheme based on the third-order TVD Runge- Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.
文摘In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.
文摘This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.
文摘In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid_layer and the reduced kinetic thickness of upper fluid_layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single_layer fluid are extended to the case of stratified fluid.
基金The work is supported by the Basic Research Foundation of the National NumericalWind Tunnel Project(Grant No.NNW2018-ZT4A08)the National Natural Science Foundation(Grant No.11972370)the National Key Project(Grant No.GJXM92579)of China.
文摘It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.
基金The authors acknowledge the partial financial support received by the E.U.through the INTAS Project 06-1000013-9236 and by the“Ministero Infrastrutture e Trasporti”within the“Programma di Ricerca 2007-2009”.Acknowledgments are also due to Prof.Maurizio Brocchini for his useful comments and suggestions.
文摘The complete analytical solution of the Riemann problem for the homo-geneous Dispersive Nonlinear Shallow Water Equations [Antuono, Liapidevskii andBrocchini, Stud. Appl. Math., 122 (2009), pp. 1-28] is presented, for both wet-bed anddry-bed conditions. Moreover, such a set of hyperbolic and dispersive depth-averagedequations shows an interesting resonance phenomenon in the wave pattern of the solu-tion and we define conditions for the occurrence of resonance and present an algorithmto capture it. As an indirect check on the analytical solution we have carried out a de-tailed comparison with the numerical solution of the government equations obtainedfrom a dissipative method that does not make explicit use of the solution of the localRiemann problem.
