A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical regime.In this paper,we introduce an efficient semi-Eulerian implementation of this me...A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical regime.In this paper,we introduce an efficient semi-Eulerian implementation of this method.The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation.The difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers.This yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase plane.As a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method.Moreover,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this paper.Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.展开更多
A semi-implicit and Eulerian - Lagrangian finite difference method for three-dimensionalshallow flow has been extended to a more complete system of equations incorporating second-moment turbulence closure model and tr...A semi-implicit and Eulerian - Lagrangian finite difference method for three-dimensionalshallow flow has been extended to a more complete system of equations incorporating second-moment turbulence closure model and transport equations of salinity and temperature. The simulation for flooding and drying of mudflats has been improved. The model is applied to Xiamen waters. Based on extensive survey data, water level elevation, temperature and salinity field along the eastern open boundary and at the Jiulong River inlets and runoffs are analyzed, specified and calibrated. The computed results show good agreement with the measured data, reproduce flooding, emergence of large and complex mudflat region.展开更多
The main river, the Dongting Lake and river networks in the Jingjiang reach of the Yangtze River constitute a complex water system, for which a full 2-D hydrodynamic model is established instead of the traditional 1-D...The main river, the Dongting Lake and river networks in the Jingjiang reach of the Yangtze River constitute a complex water system, for which a full 2-D hydrodynamic model is established instead of the traditional 1-D or compound models for simulation of such complex systems, based on the latest developments of computer technologies and numerical methods. To better handle irregular boundaries and keep the computation cost well in a reasonable limit, unstructured grids of moderate scale are used. In addition, a dynamic boundary tracking method is proposed to simulate variable flow domains at different floods, especially, when the moderate scale gird can not describe flows in narrow river-network channels at low water levels. The t9 semi-implicit method and the Eulerian-Lagrangian Method （ELM） are adopted, which make the model unconditionally stable with respect to the gravity wave speed and Courant number restrictions. Properties and efficiency of the model are discussed, and it is concluded that the new model is robust and efficient enough for the simulation of a big, complex water system. Validation tests show that the simulation results agree well with field data. It takes about 0.96 h to complete the computation of a 76 d flood, which indicates that the model is efficient enough for engineering applications.展开更多
基金This work was partially supported by NSF grant No.DMS-0608720,NSF FRG grant DMS-0757285,NSAF Projects 10676017,NSFC Projects 10971115the National Basic Research Program of China under the grant 2005CB321701.SJin was also supported by a Van Vleck Distinguished Research Prize from University of Wisconsin-Madison.
文摘A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical regime.In this paper,we introduce an efficient semi-Eulerian implementation of this method.The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation.The difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers.This yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase plane.As a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method.Moreover,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this paper.Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.
文摘A semi-implicit and Eulerian - Lagrangian finite difference method for three-dimensionalshallow flow has been extended to a more complete system of equations incorporating second-moment turbulence closure model and transport equations of salinity and temperature. The simulation for flooding and drying of mudflats has been improved. The model is applied to Xiamen waters. Based on extensive survey data, water level elevation, temperature and salinity field along the eastern open boundary and at the Jiulong River inlets and runoffs are analyzed, specified and calibrated. The computed results show good agreement with the measured data, reproduce flooding, emergence of large and complex mudflat region.
基金supported by the Eleventh"Five-Year Plan" Science and Technology Program of China(Grant No. 2008BAB29B08)the National Key Basic Research Program of China(973 Program,Grant No.2007CB714100)supported by the Yangtze River Scientific Research Institute project(Grant No.CKSQ2010075)
文摘The main river, the Dongting Lake and river networks in the Jingjiang reach of the Yangtze River constitute a complex water system, for which a full 2-D hydrodynamic model is established instead of the traditional 1-D or compound models for simulation of such complex systems, based on the latest developments of computer technologies and numerical methods. To better handle irregular boundaries and keep the computation cost well in a reasonable limit, unstructured grids of moderate scale are used. In addition, a dynamic boundary tracking method is proposed to simulate variable flow domains at different floods, especially, when the moderate scale gird can not describe flows in narrow river-network channels at low water levels. The t9 semi-implicit method and the Eulerian-Lagrangian Method （ELM） are adopted, which make the model unconditionally stable with respect to the gravity wave speed and Courant number restrictions. Properties and efficiency of the model are discussed, and it is concluded that the new model is robust and efficient enough for the simulation of a big, complex water system. Validation tests show that the simulation results agree well with field data. It takes about 0.96 h to complete the computation of a 76 d flood, which indicates that the model is efficient enough for engineering applications.
