This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation techn...This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...展开更多
In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simu...In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.展开更多
By using the generalized characteristic analysis method, the two-dimensional four-wave Riemann problem for scalar conservation laws, which is nonconvex along the y direction, was studied. Riemann solutions, which invo...By using the generalized characteristic analysis method, the two-dimensional four-wave Riemann problem for scalar conservation laws, which is nonconvex along the y direction, was studied. Riemann solutions, which involve the Guckenheimer structure, were constructed.展开更多
This paper is concerned with Godunov＇s scheme for the initial-boundary value problem of scalar conservation laws. A kind of Godunov＇s scheme, which satisfies the boundary entropy condition, was given. By use of the ...This paper is concerned with Godunov＇s scheme for the initial-boundary value problem of scalar conservation laws. A kind of Godunov＇s scheme, which satisfies the boundary entropy condition, was given. By use of the scheme, numerical simulation for the weak entropy solution to the initial-boundary value problem of scalar conservation laws is conducted.展开更多
This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. B...This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.展开更多
In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws,we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By ap...In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws,we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the conver gence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak L∞-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.展开更多
The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws.Firstly,a kinetic formulation of entropy solutions is established.Secondly,by using the kinetic formulation a...The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws.Firstly,a kinetic formulation of entropy solutions is established.Secondly,by using the kinetic formulation and kinetic techniques,the uniqueness of entropy solutions is obtained.Finally,the parabolic approximation is studied and an error estimate of order η 1/3 between the entropy solution and the viscous approximate solutions is established by using kinetic techniques,where η is the size of artificial viscosity.展开更多
Numerical solutions of Riemann problems for 2-D scalar conservation law are given by a second order accurate MmB (locally Maximum-minimum Bounds preserving) scheme which is non-splitting. The numerical computations s...Numerical solutions of Riemann problems for 2-D scalar conservation law are given by a second order accurate MmB (locally Maximum-minimum Bounds preserving) scheme which is non-splitting. The numerical computations show that the scheme has high resolution and non-oscillatory properties. The results are completely in accordance with the theoretical solutions and all cases are distinguished efficiently展开更多
This paper research on the pointwise behavior of perturbations from a viscous shock solution to a scalar viscous conservation law by introducing an approximate Green’s function. The authors obtain not only the pointw...This paper research on the pointwise behavior of perturbations from a viscous shock solution to a scalar viscous conservation law by introducing an approximate Green’s function. The authors obtain not only the pointwise decay of the perturbation and but also the high derivative of it. Stability in anyL(p≥1) norm is a direct consequence.展开更多
In this paper, we study the inviscid limit problem for the scalar viscous conservation laws on half plane. We prove that if the solution of the corresponding inviscid equation on half plane is piecewise smooth with a ...In this paper, we study the inviscid limit problem for the scalar viscous conservation laws on half plane. We prove that if the solution of the corresponding inviscid equation on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to the viscous conservation laws which converge to the inviscid solution away from the shock discontinuity and the boundary at a rate of ε1 as the viscosity ε tends to zero.展开更多
This paper deals with a scalar conservation law in 1-D space dimension, and in particular, the focus is on the stability analysis for such an equation. The problem of feedback stabilization under proportional-integral...This paper deals with a scalar conservation law in 1-D space dimension, and in particular, the focus is on the stability analysis for such an equation. The problem of feedback stabilization under proportional-integral-derivative(PID for short) boundary control is addressed. In the proportional-integral(PI for short) controller case, by spectral analysis, the authors provide a complete characterization of the set of stabilizing feedback parameters, and determine the corresponding time delay stability interval. Moreover, the stability of the equilibrium is discussed by Lyapunov function techniques, and by this approach the exponential stability when a damping term is added to the classical PI controller scheme is proved. Also, based on Pontryagin results on stability for quasipolynomials, it is shown that the closed-loop system sub ject to PID control is always unstable.展开更多
This paper is devoted to the existence and the uniqueness of the entropy solution for a general scalar conservation law associated with a forced bilateral obstacle condition in a bounded domain of Rp, p>= 1. The me...This paper is devoted to the existence and the uniqueness of the entropy solution for a general scalar conservation law associated with a forced bilateral obstacle condition in a bounded domain of Rp, p>= 1. The method of penalization is used with a view to obtaining an existence result. However, the former only gives uniform L -estimates and so leads in fact to look for an Entropy Measure-Valued Solution, according to the specific properties of bounded sequences in L . The uniqueness of this EMVS is proved. Classically, it first ensures the existence of a bounded and measurable function U entropy solution and then the strong convergence in Lq of approximate solutions to U.展开更多
基金supported by the NSF China#10571075NSF-Guangdong China#04010473+1 种基金The research of the second author was supported by Jinan University Foundation#51204033the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State education Ministry#2005-383
文摘This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...
基金Project supported by National Natural Science Foundation of China(Grant No .10271072)
文摘In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.
文摘By using the generalized characteristic analysis method, the two-dimensional four-wave Riemann problem for scalar conservation laws, which is nonconvex along the y direction, was studied. Riemann solutions, which involve the Guckenheimer structure, were constructed.
基金Project supported by the National Natural Science Foundation of China （Grant No. 10671120）
文摘This paper is concerned with Godunov＇s scheme for the initial-boundary value problem of scalar conservation laws. A kind of Godunov＇s scheme, which satisfies the boundary entropy condition, was given. By use of the scheme, numerical simulation for the weak entropy solution to the initial-boundary value problem of scalar conservation laws is conducted.
基金the NSF-Guangdong China(04010473)Jinan University Foundation(51204033)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(No.2005-383)
文摘This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.
文摘In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws,we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the conver gence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak L∞-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.
基金supported by the National Natural Science Foundation of China (No. 10971135)the Program for New Century Excellent Talents of the Ministry of Education of China (No. NCET-07-0546)+2 种基金the University Young Teacher Sciences Foundation of Anhui Province (No. 2010SQRL145)Shanghai Jiao Tong University Innovation Fund for Postgraduates (No. AE071202)the Quality Project Fund of Fuyang Teachers College (No. 2010JPKC07)
文摘The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws.Firstly,a kinetic formulation of entropy solutions is established.Secondly,by using the kinetic formulation and kinetic techniques,the uniqueness of entropy solutions is obtained.Finally,the parabolic approximation is studied and an error estimate of order η 1/3 between the entropy solution and the viscous approximate solutions is established by using kinetic techniques,where η is the size of artificial viscosity.
文摘Numerical solutions of Riemann problems for 2-D scalar conservation law are given by a second order accurate MmB (locally Maximum-minimum Bounds preserving) scheme which is non-splitting. The numerical computations show that the scheme has high resolution and non-oscillatory properties. The results are completely in accordance with the theoretical solutions and all cases are distinguished efficiently
基金the National Natural Science Foundation of China(10131050)
文摘This paper research on the pointwise behavior of perturbations from a viscous shock solution to a scalar viscous conservation law by introducing an approximate Green’s function. The authors obtain not only the pointwise decay of the perturbation and but also the high derivative of it. Stability in anyL(p≥1) norm is a direct consequence.
基金Acknowledgments The author is supported by Tianyuan Foundation （No. 11026093） and the National Natural Science Foundation of China （Nos. 11101162, 11071086）.
文摘In this paper, we study the inviscid limit problem for the scalar viscous conservation laws on half plane. We prove that if the solution of the corresponding inviscid equation on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to the viscous conservation laws which converge to the inviscid solution away from the shock discontinuity and the boundary at a rate of ε1 as the viscosity ε tends to zero.
基金supported by the ERC Advanced Grant 266907(CPDENL)of the 7th Research Framework Programme(FP7)FIRST,Initial Training Network of the European Commission(No.238702)PITNGA-2009-238702
文摘This paper deals with a scalar conservation law in 1-D space dimension, and in particular, the focus is on the stability analysis for such an equation. The problem of feedback stabilization under proportional-integral-derivative(PID for short) boundary control is addressed. In the proportional-integral(PI for short) controller case, by spectral analysis, the authors provide a complete characterization of the set of stabilizing feedback parameters, and determine the corresponding time delay stability interval. Moreover, the stability of the equilibrium is discussed by Lyapunov function techniques, and by this approach the exponential stability when a damping term is added to the classical PI controller scheme is proved. Also, based on Pontryagin results on stability for quasipolynomials, it is shown that the closed-loop system sub ject to PID control is always unstable.
文摘This paper is devoted to the existence and the uniqueness of the entropy solution for a general scalar conservation law associated with a forced bilateral obstacle condition in a bounded domain of Rp, p>= 1. The method of penalization is used with a view to obtaining an existence result. However, the former only gives uniform L -estimates and so leads in fact to look for an Entropy Measure-Valued Solution, according to the specific properties of bounded sequences in L . The uniqueness of this EMVS is proved. Classically, it first ensures the existence of a bounded and measurable function U entropy solution and then the strong convergence in Lq of approximate solutions to U.
