High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of ...High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).展开更多
In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton ...In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.展开更多
A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability o...A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.展开更多
In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equ...In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory展开更多
This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of init...This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.展开更多
High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discont...High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discontinuous Galerkin(DG) method for solving the 3D compressible Euler and Navier-Stokes equations. In this method, a fine linear mesh is first generated by standard commercial mesh generation tools. By taking advantage of an agglomeration method, a quadratic high-order mesh is quickly obtained, which is coarse but provides a high-quality geometry representation, thus very suitable for high-order computations. High-order discretizations are performed on the obtained grids with DG method and the discretized system is treated fully implicitly to obtain steady state solutions. Numerical experiments on several flow problems indicate that the agglomerated high-order mesh works well with DG method in dealing with flow problems of curved geometries. It is also found that with a fully implicit discretized system and a p-sequencing method, the DG method can achieve convergence state within several time steps which shows significant efficiency improvements compared to its explicit counterparts.展开更多
In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing...In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.展开更多
We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equa...We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.展开更多
The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,wh...The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.展开更多
In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtain...In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkinmethod recently proposed in[20].In the Lagrangian framework considered here,the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element.For the spacetime basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points.The moving space-time elements are mapped to a reference element using an isoparametric approach,i.e.the spacetime mapping is defined by the same basis functions as the weak solution of the PDE.We show some computational examples in one space-dimension for non-stiff and for stiff balance laws,in particular for the Euler equations of compressible gas dynamics,for the resistive relativistic MHD equations,and for the relativistic radiation hydrodynamics equations.Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.展开更多
文摘High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).
文摘In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.
基金J.Qiu is supported partly by National Natural Science Foundation(China)grant 12071392.
文摘A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.
基金supported by NSFC 40906048.The research of J.Qiu was supported by NSFC 10671091 and 10811120283support was provided by USA NSF DMS-0820348 while he was in residence at Department of Mathematical Sciences,Rensselaer Polytechnic Institutesupported by NSF of Hohai University 2048/408306
文摘In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory
文摘This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.
基金co-supported by the Aeronautical Science Foundation of China (No. 20152752033)the National Natural Science Foundation of China (No. 11272152)the Open Project of Key Laboratory of Aerodynamic Noise Control
文摘High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discontinuous Galerkin(DG) method for solving the 3D compressible Euler and Navier-Stokes equations. In this method, a fine linear mesh is first generated by standard commercial mesh generation tools. By taking advantage of an agglomeration method, a quadratic high-order mesh is quickly obtained, which is coarse but provides a high-quality geometry representation, thus very suitable for high-order computations. High-order discretizations are performed on the obtained grids with DG method and the discretized system is treated fully implicitly to obtain steady state solutions. Numerical experiments on several flow problems indicate that the agglomerated high-order mesh works well with DG method in dealing with flow problems of curved geometries. It is also found that with a fully implicit discretized system and a p-sequencing method, the DG method can achieve convergence state within several time steps which shows significant efficiency improvements compared to its explicit counterparts.
基金This work is supported by the NNSFC （Nos. 11771213, 41504078, 11301234, 11271171）, the National Key Research and Development Project of China （No. 2016YFC0600310）, the Major Projects of Natural Sciences of University in Jiangsu Province of China （No. 15KJA110002） and the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Provincial Natural Science Foundation of Jiangxi （Nos. 20161ACB20006, 20142BCB23009, 20151BAB 201012）.
文摘In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.
基金supported by National Science Foundation of China(52171251)Liao Ning Revitalization Talents Program(XLYC1907014)+2 种基金the Fundamental Research Funds for the Central Universities(DUT21ZD205)Ministry of Industry and Information Technology(2019-357)the Project of State Key Laboratory of Satellite Ocean Environment Dynamics,Second Institute of Oceanography,MNR(QNHX2112)。
文摘We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.
基金the National Numerical Wind Tunnel Project(No.NNW2018-ZT4A08)the National Natural Science Foundation of China(Nos.11872323 and 11971025)the Natural Science Foundation of Fujian Province(No.2019J06002)。
文摘The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.
基金the European Research Council under the European Union’s Seventh Framework Programme(FP7/2007-2013)under the research project STiMulUs,ERC Grant agreement no.278267.
文摘In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkinmethod recently proposed in[20].In the Lagrangian framework considered here,the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element.For the spacetime basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points.The moving space-time elements are mapped to a reference element using an isoparametric approach,i.e.the spacetime mapping is defined by the same basis functions as the weak solution of the PDE.We show some computational examples in one space-dimension for non-stiff and for stiff balance laws,in particular for the Euler equations of compressible gas dynamics,for the resistive relativistic MHD equations,and for the relativistic radiation hydrodynamics equations.Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.
