In this paper, we propose a numerical method based on semi-Lagrangian approach for solving quasi-geostrophic (QG) equations on a sphere. Using potential vorticity and stream-function as prognostic variables, two-...In this paper, we propose a numerical method based on semi-Lagrangian approach for solving quasi-geostrophic (QG) equations on a sphere. Using potential vorticity and stream-function as prognostic variables, two-order centered difference is suggested on the latitude-longitude grid. In our proposed numerical scheme, advection terms are expressed in a Lagrangian frame of reference to circumvent the CFL restriction. The pole singularity associated with the latitude-longitude grid is eliminated by a smoothing technique for the initial flow. Error analysis is provided for the numerical scheme.展开更多
Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interface...Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed.Note that,while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature,there is a gap in dealing with Navier-Stokes equations,where the non-linear term includes an additional component involving the coordinateφ,even if the geometry and force term are axis-symmetric.Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity,in addition to the pole singularity at r=0.In the newly developed algorithm,we have derived the jump conditions under the cylindrical coordinates.The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates.The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface.As expected,the method is shown to be second-order accurate for the velocity.The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.展开更多
文摘In this paper, we propose a numerical method based on semi-Lagrangian approach for solving quasi-geostrophic (QG) equations on a sphere. Using potential vorticity and stream-function as prognostic variables, two-order centered difference is suggested on the latitude-longitude grid. In our proposed numerical scheme, advection terms are expressed in a Lagrangian frame of reference to circumvent the CFL restriction. The pole singularity associated with the latitude-longitude grid is eliminated by a smoothing technique for the initial flow. Error analysis is provided for the numerical scheme.
基金supported by Fundación Sáneca Grant 21728/EE/22(Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad,Colaboración Internacional e Intercambio de Conocimiento“Jiménez de la Espada”)B.Dong is partially supported by the National Natural Science Foundation of China Grant No.12261070+1 种基金Ningxia Key Research and Development Project of China Grant No.2022BSB03048Z.Li is partially supported by Simons Grant 633724 and by Fundación Séneca Grant 21760/IV/22(Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad,Colaboración Internacional Intercambio de Conocimiento“Jiménez de la Espada”).
文摘Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed.Note that,while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature,there is a gap in dealing with Navier-Stokes equations,where the non-linear term includes an additional component involving the coordinateφ,even if the geometry and force term are axis-symmetric.Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity,in addition to the pole singularity at r=0.In the newly developed algorithm,we have derived the jump conditions under the cylindrical coordinates.The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates.The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface.As expected,the method is shown to be second-order accurate for the velocity.The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.
