The authors consider systems of the form where the matrix A(u) is assumed to be strictly hyperbolic and with the property that the integral curves of the eigenvector fields are straight lines. For this class of system...The authors consider systems of the form where the matrix A(u) is assumed to be strictly hyperbolic and with the property that the integral curves of the eigenvector fields are straight lines. For this class of systems one can define a natural Riemann solver, and hence a Godunov scheme, which generalize the standard Riemann solver and Godunov scheme for conservative systems. This paper shows convergence and L1 stability for this scheme when applied to data with small total variation. The main step in the proof is to estimate the increase in the total variation produced by the scheme due to quadratic coupling terms. Using Duhamel’s principle, the problem is reduced to the estimate of the product of two Green kernels, representing probability densities of discrete random walks. The total amount of coupling is then determined by the expected number of crossings between two random walks with strictly different average speeds. This provides a discrete analogue of the arguments developed in [3,9] in connection with continuous random processes.展开更多
The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obta...The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.展开更多
This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-sol...This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.展开更多
In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficien...In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given.展开更多
This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 p...This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.展开更多
The following nonlinear hyperbolic equation is discussed in this paper:where A= -? + Iandx∈Rn. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local ...The following nonlinear hyperbolic equation is discussed in this paper:where A= -? + Iandx∈Rn. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.展开更多
Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are pro...Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.展开更多
Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the itera...Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.展开更多
This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local so...This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local solution of the problem.展开更多
In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing ac...In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.展开更多
This paper is concerned with a numerical solution of hyperbolic cooling tower shell, a class of full nonlinear problems in solid mechanics of considerable interest in engineering applications. In this analysis, the po...This paper is concerned with a numerical solution of hyperbolic cooling tower shell, a class of full nonlinear problems in solid mechanics of considerable interest in engineering applications. In this analysis, the post-buckling analysis of cooling tower shell with discrete fixed support and under the action of wind loads and dead load is studied. The influences of ring-stiffener on instability load are also discussed. In addition, a new solution procedure for nonlinear problems which is the combination of load increment iteration with modified R-C are-length method is suggested. Finally, some conclusions having important significance for practice engineering are given.展开更多
A nonlinear damped system with boundary input and output, which also has source term, is studied in this paper. It is proved that under some conditions the system has global solution and blow-up solution.
In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obta...In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obtain a global Holder continuous solution and the convergent rate of the viscosity method for the Cauchy problem of the variant nonisentropic system of polytropic gas for any adiabatic exponentγ>1.展开更多
The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a ...The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a 2u xxt -a 3u xxtt =φ(u x ) x are proved.展开更多
This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms.A Crank\|Nicolson approximation for this kind of equations is presented.By using...This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms.A Crank\|Nicolson approximation for this kind of equations is presented.By using the elliptic Ritz\|Volterra projection,the analysis of the error estimates for the finite element numerical solutions and the optimal H \+1\|norm error estimate are demonstrated.展开更多
In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enj...In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enjoys maximum principle, monotonicity-preserving and entropy condition on the continuum level. Moreover, it has a well-defined local limit given by a conventional local conservation laws in the form of partial differential equations. We discuss convergent numerical approximations that preserve similar properties on the discrete level.We also present numerical experiments to study various limiting behavior of the numerical solutions.展开更多
We consider strictly hyperbolic nonlinear equations which are Lipschitz continuous in the time variable and study the local analytic regularity of the solutions with respect to the space variables.
A PLU-SGS method based on a time-derivative preconditioning algorithm and LU-SGS method is developed in order to calculate the Navier-Stokes equations at all speeds. The equations were discretized using A USMPW scheme...A PLU-SGS method based on a time-derivative preconditioning algorithm and LU-SGS method is developed in order to calculate the Navier-Stokes equations at all speeds. The equations were discretized using A USMPW scheme in conjunction with the third-order MUSCL scheme with Van Leer limiter. The present method was applied to solve the multidimensional compressible Navier-Stokes equations in curvilinear coordinates. Characteristic boundary conditions based on the eigensystem of the preconditioned equations were employed. In order to examine the performance of present method, driven-cavity flow at various Reynolds numbers and viscous flow through a convergent-divergent nozzle at supersonic were selected to rest this method. The computed results were compared with the experimental data or the other numerical results available in literature and good agreements between them are obtained. The results show that the present method is accurate, self-adaptive and stable for a wide range of flow conditions from low speed to supersonic flows.展开更多
A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value probl...A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value problems.展开更多
基金the European TMR network"Hyperbolic Systems of Conservation Laws"! ERBFMRXCT960033
文摘The authors consider systems of the form where the matrix A(u) is assumed to be strictly hyperbolic and with the property that the integral curves of the eigenvector fields are straight lines. For this class of systems one can define a natural Riemann solver, and hence a Godunov scheme, which generalize the standard Riemann solver and Godunov scheme for conservative systems. This paper shows convergence and L1 stability for this scheme when applied to data with small total variation. The main step in the proof is to estimate the increase in the total variation produced by the scheme due to quadratic coupling terms. Using Duhamel’s principle, the problem is reduced to the estimate of the product of two Green kernels, representing probability densities of discrete random walks. The total amount of coupling is then determined by the expected number of crossings between two random walks with strictly different average speeds. This provides a discrete analogue of the arguments developed in [3,9] in connection with continuous random processes.
基金supported by National Natural Science Foundation of China(61273016)The Natural Science Foundation of Zhejiang Province(Y6100016)The Public Welfare Technology Application Research Project of Zhejiang Province Science and Technology Department(2015C33088)
文摘The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.
基金Project supported by the NSF of Fujian Province (A97020)
文摘This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.
文摘In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given.
基金Supported by the National Natural Science Foundation of China (10671184)
文摘This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.
文摘The following nonlinear hyperbolic equation is discussed in this paper:where A= -? + Iandx∈Rn. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.
基金Project supported by the National Natural Science Foundation of China (Nos. 10371073 and 10572156) the Natural Science Foundation of Henan Province of China (No.0611050500)
文摘Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.
文摘Presents the iterative method of solving Cauchy problem with reproducing kernel for nonlinear hyperbolic equations, and the application of the computational technique of reproducing kernel space to simplify, the iterative computation and increase the convergence rate and points out that this method is still effective. Even if the initial condition is discrete.
基金Supported by the National Natural Science Foundation of China（10671182） Supported by the Excellent Youth Teachers Foundation of High College of Henan Province（2006110016）
文摘This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local solution of the problem.
基金the fellowship of China Postdoctoral Science Foundation,no:2020TQ0030.Y.Xu:Research supported by National Numerical Windtunnel Project NNW2019ZT4-B08+1 种基金Science Challenge Project TZZT2019-A2.3NSFC Grants 11722112,12071455.X.Li:Research supported by NSFC Grant 11801062.
文摘In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.
基金Project Supported by National Natural Science Foundation of China
文摘This paper is concerned with a numerical solution of hyperbolic cooling tower shell, a class of full nonlinear problems in solid mechanics of considerable interest in engineering applications. In this analysis, the post-buckling analysis of cooling tower shell with discrete fixed support and under the action of wind loads and dead load is studied. The influences of ring-stiffener on instability load are also discussed. In addition, a new solution procedure for nonlinear problems which is the combination of load increment iteration with modified R-C are-length method is suggested. Finally, some conclusions having important significance for practice engineering are given.
基金Supported by the National Science Foundation of China (60774014)the Science Foundation of Shanxi Province (2007011002)
文摘A nonlinear damped system with boundary input and output, which also has source term, is studied in this paper. It is proved that under some conditions the system has global solution and blow-up solution.
基金supported by the National Natural Science Foundation of China(12071409)。
文摘In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obtain a global Holder continuous solution and the convergent rate of the viscosity method for the Cauchy problem of the variant nonisentropic system of polytropic gas for any adiabatic exponentγ>1.
基金the National Natural Science Foundation of China(1 0 0 71 0 74) and the Natural ScienceFoundation of Henan Provinc
文摘The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a 2u xxt -a 3u xxtt =φ(u x ) x are proved.
文摘This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms.A Crank\|Nicolson approximation for this kind of equations is presented.By using the elliptic Ritz\|Volterra projection,the analysis of the error estimates for the finite element numerical solutions and the optimal H \+1\|norm error estimate are demonstrated.
基金Supported in part by the NSF(Grant No.DMS-1558744)the AFOSR MURI Center for Material Failure Prediction Through Peridynamics and the ARO MURI(Grant No.W911NF-15-1-0562)
文摘In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enjoys maximum principle, monotonicity-preserving and entropy condition on the continuum level. Moreover, it has a well-defined local limit given by a conventional local conservation laws in the form of partial differential equations. We discuss convergent numerical approximations that preserve similar properties on the discrete level.We also present numerical experiments to study various limiting behavior of the numerical solutions.
文摘We consider strictly hyperbolic nonlinear equations which are Lipschitz continuous in the time variable and study the local analytic regularity of the solutions with respect to the space variables.
文摘A PLU-SGS method based on a time-derivative preconditioning algorithm and LU-SGS method is developed in order to calculate the Navier-Stokes equations at all speeds. The equations were discretized using A USMPW scheme in conjunction with the third-order MUSCL scheme with Van Leer limiter. The present method was applied to solve the multidimensional compressible Navier-Stokes equations in curvilinear coordinates. Characteristic boundary conditions based on the eigensystem of the preconditioned equations were employed. In order to examine the performance of present method, driven-cavity flow at various Reynolds numbers and viscous flow through a convergent-divergent nozzle at supersonic were selected to rest this method. The computed results were compared with the experimental data or the other numerical results available in literature and good agreements between them are obtained. The results show that the present method is accurate, self-adaptive and stable for a wide range of flow conditions from low speed to supersonic flows.
基金The project is supported by the National Natural Science Foundation of China(10071048)
文摘A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value problems.
