This paper aims to present a new well-balanced,accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws.It is a scheme that combines both finite volume approach and characteri...This paper aims to present a new well-balanced,accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws.It is a scheme that combines both finite volume approach and characteristic method.In this study,we consider a shallow water system with Coriolis effect and bottom friction stresses where this new Finite Volume Characteristics(FVC)scheme has been applied.The physical and mathematical properties of the system,including the C-property,have been well preserved.First,we developed this approach by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics,in order to avoid Riemann solvers and to enhance the accuracy without any complexity of the MUSCL reconstruction.Afterward,a discretization was applied to the bottom source term that leads to a well-balanced scheme satisfying the steady-state condition of still water.A semi-implicit treatment will also be presented in this study to avoid stability problems due to source terms.Finally,the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results;moreover,it gives a noteworthy accuracy and rapidity improvement compared to the original approaches.展开更多
We propose a new method for numerical solution of the third-order differential equations.The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear sec...We propose a new method for numerical solution of the third-order differential equations.The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter.The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations.A non-oscillatory finite volume method for the relaxation system is developed.The method is uniformly accurate for all relaxation rates.Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation.Our method demonstrated the capability of accurately capturing soliton wave phenomena.展开更多
The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension.The governing equations consist of two components,namely a hydra...The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension.The governing equations consist of two components,namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation.Based on different formulations of the morphodynamic equations,we propose a family of three finite volume methods.The numerical fluxes are reconstructed using a modified Roe’s scheme that incorporates,in its reconstruction,the sign of the Jacobian matrix in the morphodynamic system.A well-balanced discretization is used for the treatment of the source terms.The method is wellbalanced,non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport.The obtained results for several morphodynamic problems are considered to be representative,and might be helpful for a fair rating of finite volume solution schemes,particularly in long time computations.展开更多
基金supported by the HPC Project Alkhwarizmi department,MSDA-UM6P.
文摘This paper aims to present a new well-balanced,accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws.It is a scheme that combines both finite volume approach and characteristic method.In this study,we consider a shallow water system with Coriolis effect and bottom friction stresses where this new Finite Volume Characteristics(FVC)scheme has been applied.The physical and mathematical properties of the system,including the C-property,have been well preserved.First,we developed this approach by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics,in order to avoid Riemann solvers and to enhance the accuracy without any complexity of the MUSCL reconstruction.Afterward,a discretization was applied to the bottom source term that leads to a well-balanced scheme satisfying the steady-state condition of still water.A semi-implicit treatment will also be presented in this study to avoid stability problems due to source terms.Finally,the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results;moreover,it gives a noteworthy accuracy and rapidity improvement compared to the original approaches.
文摘We propose a new method for numerical solution of the third-order differential equations.The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter.The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations.A non-oscillatory finite volume method for the relaxation system is developed.The method is uniformly accurate for all relaxation rates.Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation.Our method demonstrated the capability of accurately capturing soliton wave phenomena.
基金This work was partly performed while the third author was a visiting professor at CMLA,Ecole normale superieure de Cachan.Financial support provided by CLMA,ENS Cachan is gratefully acknowledged。
文摘The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension.The governing equations consist of two components,namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation.Based on different formulations of the morphodynamic equations,we propose a family of three finite volume methods.The numerical fluxes are reconstructed using a modified Roe’s scheme that incorporates,in its reconstruction,the sign of the Jacobian matrix in the morphodynamic system.A well-balanced discretization is used for the treatment of the source terms.The method is wellbalanced,non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport.The obtained results for several morphodynamic problems are considered to be representative,and might be helpful for a fair rating of finite volume solution schemes,particularly in long time computations.
