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 the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a con...This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of 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 global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.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
In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma...In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*.展开更多
This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um ar...This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.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 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 the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first par...In the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first part of the series[6],the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly.By applying this technique,the authors demonstrate that a pure continu-ous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way.In this work,we extend this investigation to the nonlinear case and focus on entropy conservation.By switching to entropy variables,we provide an estimation of the boundary operators also for nonlinear problems,that guarantee conservation.In numerical simulations,we verify our theoretical analysis.展开更多
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.
文摘This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of 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 global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.
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supported in part by NSFC(11671193)the Fundamental Research Funds for the Central Universities(NE2015005)supported in part by NSFC(11271182 and 11501290)
文摘In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*.
文摘This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.
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 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｜）.
基金funded by the SNF Grant(Number 200021175784)the UZH Postdoc grant+1 种基金funded by an SNF Grant 200021_153604The Los Alamos unlimited release number is LA-UR-19-32411.
文摘In the hyperbolic research community,there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability.In the first part of the series[6],the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly.By applying this technique,the authors demonstrate that a pure continu-ous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way.In this work,we extend this investigation to the nonlinear case and focus on entropy conservation.By switching to entropy variables,we provide an estimation of the boundary operators also for nonlinear problems,that guarantee conservation.In numerical simulations,we verify our theoretical analysis.
