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
A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,w...A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,which simplifies the computation of suitable coupling data.The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit.The role of this condition is discussed,a consistency concept with respect to the original problem is introduced,the well-posedness is analyzed and explicit,nodal Riemann solvers are provided.Based on a case study considering the p-system of gas dynamics,a strategy for the design of the relaxation coupling condition within the new scheme is provided.展开更多
This paper continues to study the central relaxing schemes for system of hyperbolic conservation laws, based on the local relaxation approximation. Two classes of relaxing systems with stiff source term are introduced...This paper continues to study the central relaxing schemes for system of hyperbolic conservation laws, based on the local relaxation approximation. Two classes of relaxing systems with stiff source term are introduced to approximate system of conservation laws in curvilinear coordinates. Based on them, the semi-implicit relaxing schemes are con- structed as in [6, 12] without using any linear or nonlinear Riemann solvers. Numerical experiments for one-dimensional and two-dimensional problems are presented to demon- strate the performance and resolution of the current schemes.展开更多
In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type centr...In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type central scheme,the CWENOtype central-upwind scheme,i.e.,the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction,is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies.The high resolution,the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.展开更多
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
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.
基金Funding Open Access funding enabled and organized by Projekt DEALthe Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)for the financial support through 320021702/GRK2326,333849990/IRTG-2379,B04,B05,and B06 of 442047500/SFB1481,HE5386/18-1,19-2,22-1,23-1,25-1,ERS SFDdM035 and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612 and under the Excellence Strategy of the Federal Government and the Länder.Support through the EU DATAHYKING is also acknowledged.
文摘A novel numerical scheme to solve two coupled systems of conservation laws is introduced.The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems,which simplifies the computation of suitable coupling data.The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit.The role of this condition is discussed,a consistency concept with respect to the original problem is introduced,the well-posedness is analyzed and explicit,nodal Riemann solvers are provided.Based on a case study considering the p-system of gas dynamics,a strategy for the design of the relaxation coupling condition within the new scheme is provided.
基金This project supported partly by National Natural Science Foundation of China (No.19901031), the specialFunds for Major State
文摘This paper continues to study the central relaxing schemes for system of hyperbolic conservation laws, based on the local relaxation approximation. Two classes of relaxing systems with stiff source term are introduced to approximate system of conservation laws in curvilinear coordinates. Based on them, the semi-implicit relaxing schemes are con- structed as in [6, 12] without using any linear or nonlinear Riemann solvers. Numerical experiments for one-dimensional and two-dimensional problems are presented to demon- strate the performance and resolution of the current schemes.
基金supported by the National Natural Science Foundation of China(Grant Nos.11101333,11071196,11171043)the National Natural Science Foundation of Shaanxi(Grant No.2011GQ1018)NPU Foundation for Fundamental Research.
文摘In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type central scheme,the CWENOtype central-upwind scheme,i.e.,the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction,is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies.The high resolution,the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.
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.
