Hamilton-Jacobi equation appears frequently in applications, e.g., in differential games and control theory, and is closely related to hyperbolic conservation laws[3, 4, 12]. This is helpful in the design of differenc...Hamilton-Jacobi equation appears frequently in applications, e.g., in differential games and control theory, and is closely related to hyperbolic conservation laws[3, 4, 12]. This is helpful in the design of difference approximations for Hamilton-Jacobi equation and hyperbolic conservation laws. In this paper we present the relaxing system for HamiltonJacobi equations in arbitrary space dimensions, and high resolution relaxing schemes for Hamilton-Jacobi equation, based on using the local relaxation approximation. The schemes are numerically tested on a variety of 1D and 2D problems, including a problem related to optimal control problem. High-order accuracy in smooth regions, good resolution of discontinuities, and convergence to viscosity solutions are observed.展开更多
We study the Green＇s function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general s...We study the Green＇s function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general systems under the basic Kawashima- Shizuta type of conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves pertaining to different characteristic families. The second result is on the continuous differentiability of the Green＇s function with respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.展开更多
The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of r...The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper.展开更多
In this paper we study the problem of the global existence （in time） of weak entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models th...In this paper we study the problem of the global existence （in time） of weak entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit （large） threshold, then the Cauchy problem has global solutions.展开更多
A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CW...A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.展开更多
基金the National Natural Science Foundation of China (Grant No. 19901031)and the foundation of National Laboratory of Computationa
文摘Hamilton-Jacobi equation appears frequently in applications, e.g., in differential games and control theory, and is closely related to hyperbolic conservation laws[3, 4, 12]. This is helpful in the design of difference approximations for Hamilton-Jacobi equation and hyperbolic conservation laws. In this paper we present the relaxing system for HamiltonJacobi equations in arbitrary space dimensions, and high resolution relaxing schemes for Hamilton-Jacobi equation, based on using the local relaxation approximation. The schemes are numerically tested on a variety of 1D and 2D problems, including a problem related to optimal control problem. High-order accuracy in smooth regions, good resolution of discontinuities, and convergence to viscosity solutions are observed.
基金The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248The research of the second author was partially supported byNSF Grant DMS-0207154
文摘We study the Green＇s function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general systems under the basic Kawashima- Shizuta type of conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves pertaining to different characteristic families. The second result is on the continuous differentiability of the Green＇s function with respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.
文摘The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper.
文摘In this paper we study the problem of the global existence （in time） of weak entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit （large） threshold, then the Cauchy problem has global solutions.
基金the National Natural Science Foundation of China (60134010)The English text was polished by Yunming Chen.
文摘A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.
