In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the targe...In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.展开更多
The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part o...The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.展开更多
The semi-Lagrangian advection scheme is implemented on a new quasi-uniform overset （Yin-Yang） grid on the sphere. The Yin-Yang grid is a newly developed grid system in spherical geometry with two perpendicularly-ori...The semi-Lagrangian advection scheme is implemented on a new quasi-uniform overset （Yin-Yang） grid on the sphere. The Yin-Yang grid is a newly developed grid system in spherical geometry with two perpendicularly-oriented latitude-longitude grid components （called Yin and Yang respectively） that overlapp each other, and this effectively avoids the coordinate singularity and the grid convergence near the poles. In this overset grid, the way of transferring data between the Yin and Yang components is the key to maintaining the accuracy and robustness in numerical solutions. A numerical interpolation for boundary data exchange, which maintains the accuracy of the original advection scheme and is computationally efficient, is given in this paper. A standard test of the solid-body advection proposed by Williamson is carried out on the Yin-Yang grid. Numerical results show that the quasi-uniform Yin-Yang grid can get around the problems near the poles, and the numerical accuracy in the original semi-Lagrangian scheme is effectively maintained in the Yin-Yang grid.展开更多
ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compre...ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpola tion （referred to as the SL_CL scheme）. The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich＇s semi-Lagrangian advection scheme （referred to as the R2007 scheme）, the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re..R2007 scheme and the SL_CL scheme. The numerical results showed that： （1） the damping effects were remarkably reduced with the Re_R2007 scheme; and （2） the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re..R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with non uniform grid cells, which also verified the Re_R2007 scheme＇s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.展开更多
A new non-interpolating semi-Lagrangian scheme has been proposed, which can eliminate any interpolation,and consequently numerical smoothing of forecast fields. Here the new scheme is applied to KdV equation and its p...A new non-interpolating semi-Lagrangian scheme has been proposed, which can eliminate any interpolation,and consequently numerical smoothing of forecast fields. Here the new scheme is applied to KdV equation and its performance is assessed by comparing the numerical results with those produced by Ritchie＇s scheme (1986).The comparison shows that the non-interpolating semi-Lagrangian scheme appears to have efficiency advantages.展开更多
We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equati...We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.展开更多
A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid.Two kinds of moments(i.e.,point values(PV moment) at cell interface...A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid.Two kinds of moments(i.e.,point values(PV moment) at cell interfaces and volume integrated average(VIA moment) value) are defined within a single cell.The PV moment is updated by a conventional semi-Lagrangian method,while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation.Different from the spatial approximation used in the CSL2(conservative semi-Lagrangian scheme with second order polynomial function) scheme,a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment.To achieve exactly positive-definite preserving,two kinds of corrections are made on the original conservative semi-Lagrangian with rational function(CSLR)scheme.The resulting scheme is inherently conservative,non-negative,and allows a Courant number larger than one.Moreover,the spatial reconstruction can be performed within a single cell,which is very efficient and economical for practical implementation.In addition,a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry,which benefitsthe implementation of the 1 D CSLR solver with large Courant number.The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry.Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical nonnegativity,and it has the potential to transport the tracers accurately in a real atmospheric model.展开更多
The interpolation method in a semi-Lagrangian scheme is decisive to its performance. Given the number of grid points one is considering to use for the interpolation, it does not necessarily follow that maximum formal ...The interpolation method in a semi-Lagrangian scheme is decisive to its performance. Given the number of grid points one is considering to use for the interpolation, it does not necessarily follow that maximum formal accuracy should give the best results. For the advection equation, the driving force of this method is the method of the characteristics, which accounts for the flow of information in the model equation. This leads naturally to an interpolation problem since the foot point is not in general located on a grid point. We use another interpolation scheme that will allow achieving the high order for the box initial condition.展开更多
The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux...The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux in time into the integration in space. Compared with the traditional semi-Lagrange scheme, the scheme devised here tries to directly evaluate the average fluxes along cell edges. It is this difference that makes the scheme in this paper simple to implement and easily extend to nonlinear cases. The procedure of evaluation of the average fluxes only depends on the high-order spatial interpolation. Hence the scheme can be implemented as long as the spatial interpolation is available, and no additional temporal discretization is needed. In this paper, the high-order spatial discretization is chosen to be the classical 5th-order weighted essentially non-oscillatory spatial interpolation. In the end, 1D and 2D numerical results show that this method is rather robust. In addition, to exhibit the numerical resolution and efficiency of the proposed scheme, the numerical solutions of the classical 5th-order WENO scheme combined with the 3rd-order Runge-Kutta temporal discretization (WENOJS) are chosen as the reference. We find that the scheme proposed in the paper generates comparable solutions with that of WENOJS, but with less CPU time.展开更多
The paper is devised to combine the approximated semi-Lagrange weighted essentially non-oscillatory scheme and flux vector splitting. The approximated finite volume semi-Lagrange that is weighted essentially non-oscil...The paper is devised to combine the approximated semi-Lagrange weighted essentially non-oscillatory scheme and flux vector splitting. The approximated finite volume semi-Lagrange that is weighted essentially non-oscillatory scheme with Roe flux had been proposed. The methods using Roe speed to construct the flux probably generates entropy-violating solutions. More seriously, the methods maybe perform numerical instability in two-dimensional cases. A robust and simply remedy is to use a global flux splitting to substitute Roe flux. The combination is tested by several numerical examples. In addition, the comparisons of computing time and resolution between the classical weighted essentially non-oscillatory scheme (WENOJS-LF) and the semi-Lagrange weighted essentially non-oscillatory scheme (WENOEL-LF) which is presented (both combining with the flux vector splitting).展开更多
A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.Th...A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.The staggered DG scheme defines the discrete pressure on the primal triangular mesh,while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh.In this paper,a new pair of equal-order-interpolation velocity-pressure finite elements is proposed.On the primary triangular mesh(the pressure elements),the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle.On the dual mesh instead(the velocity elements),the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries,while they are continuous inside.In other words,the basis functions on the dual mesh arc built by continuous finite elements on the subtriangles.This choice allows the construction of an efficient,quadrature-free and memory saving algorithm.In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations,the arbitrary high order of accuracy in time is achieved through tire use of time-dependent test and basis functions,in combination with simple and efficient Picard iterations.Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes,not only from a computer memory point of view,but also concerning the computational time.展开更多
The Global/Regional Assimilation and PrEdiction System(GRAPES) is a newly developed global non-hydrostatic numerical prediction model,which will become the next generation medium-range opera-tional model at China Mete...The Global/Regional Assimilation and PrEdiction System(GRAPES) is a newly developed global non-hydrostatic numerical prediction model,which will become the next generation medium-range opera-tional model at China Meteorological Administration(CMA).The dynamic framework of GRAPES is featuring with fully compressible equations,nonhydrostatic or hydrostatic optionally,two-level time semi-Lagrangian and semi-implicit time integration,Charney-Phillips vertical staggering,and complex three-dimensional pre-conditioned Helmholtz solver,etc.Concerning the singularity of horizontal momentum equations at the poles,the polar discretization schemes are described,which include adoption of Arakawa C horizontal grid with ν at poles,incorporation of polar filtering to maintain the computational stability,the correction to Helmholtz equation near the poles,as well as the treatment of semi-Lagrangian interpolation to improve the departure point accuracy,etc.The balanced flow tests validate the rationality of the treatment of semi-Lagrangian departure point calculation and the polar discretization during long time integration.Held and Suarez tests show that the conservation proper-ties of GRAPES model are quite good.展开更多
The paper explores the gravity-driven flow of the thin film of a viscoelastic-fluid-based nanofluids(VFBN)along an inclined plane under non-isothermal conditions and subjected to convective cooling at the free-surface...The paper explores the gravity-driven flow of the thin film of a viscoelastic-fluid-based nanofluids(VFBN)along an inclined plane under non-isothermal conditions and subjected to convective cooling at the free-surface.The Newton’s law of cooling is used to model the convective heat-exchange with the ambient at the free-surface.The Giesekus viscoelastic constitutive model,with appropriate modifications to account for non-isothermal effects,is employed to describe the polymeric effects.The unsteady and coupled non-linear partial differential equations(PDEs)describing the model problem are obtained and solved via efficient semi-implicit numerical schemes based on finite difference methods(FDM)implemented in Matlab.The response of the VFBN velocity,temperature,thermal-conductivity and polymeric-stresses to variations in the volume-fraction of embedded nanoparticles is investigated.It is shown that these quantities all increase as the nanoparticle volume-fraction becomes higher.展开更多
A 3D dynamic core of the non-hydrostatic model GRAPES（Global/Regional Assimilation and Prediction System） is developed on the Yin-Yang grid to address the polar problem and to enhance the computational efficiency. T...A 3D dynamic core of the non-hydrostatic model GRAPES（Global/Regional Assimilation and Prediction System） is developed on the Yin-Yang grid to address the polar problem and to enhance the computational efficiency. Three-dimensional Coriolis forcing is introduced to the new core, and full representation of the Coriolis forcing makes it straightforward to share code between the Yin and Yang subdomains. Similar to that in the original GRAPES model, a semi-implicit semi-Lagrangian scheme is adopted for temporal integration and advection with additional arrangement for cross-boundary transport. Under a non-centered second-order temporal and spatial discretization, the dry nonhydrostatic frame is summarized as the solution of an elliptical problem. The resulting Helmholtz equation is solved with the Generalized Conjugate Residual solver in cooperation with the classic Schwarz method. Even though the coefficients of the equation are quite different from those in the original model, the computational procedure of the new core is just the same. The bi-cubic Lagrangian interpolation serves to provide Dirichlet-type boundary conditions with data transfer between the subdomains. The dry core is evaluated with several benchmark test cases, and all the tests display reasonable numerical stability and computing performance. Persistency of the balanced flow and development of both the mountain-induced Rossby wave and Rossby–Haurwitz wave confirms the appropriate installation of the 3D Coriolis terms in the semi-implicit semi-Lagrangian dynamic core on the Yin-Yang grid.展开更多
The recent progresses in the research and development of (NWP) in China are reviewed in this paper. The most impressive achievements are the development of direct assimilation of satellite irradiances with a 3DVAR (th...The recent progresses in the research and development of (NWP) in China are reviewed in this paper. The most impressive achievements are the development of direct assimilation of satellite irradiances with a 3DVAR (three-dimentional variational) data assimilation system and a non-hydrostatic modei with a semi-Lagrangian semi-implicit scheme. Progresses have also been made in modei physics and modei application to precipitation and environmental forecasts. Some scientific issues of great importance for further development are discussed.展开更多
Quantitative studies of scientific problems require solving correspondent mathematical models. Although a great deal of mathematical models of evolutional problems are set up under continuous space-time meaning, they ...Quantitative studies of scientific problems require solving correspondent mathematical models. Although a great deal of mathematical models of evolutional problems are set up under continuous space-time meaning, they usually had to be solved numerically after space-time discretization because nonlinear mathematical models except some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
基金AFOSR and NSF for their support of this work under grants FA9550-19-1-0281 and FA9550-17-1-0394 and NSF grant DMS 191218。
文摘In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.
文摘The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.
基金This paper is sponsored by the National Natural Science Foundation of China （No. 40575050） the National Key Program for Developing Basic Reseach （＂973＂） （No. 2004CB418306）.
文摘The semi-Lagrangian advection scheme is implemented on a new quasi-uniform overset （Yin-Yang） grid on the sphere. The Yin-Yang grid is a newly developed grid system in spherical geometry with two perpendicularly-oriented latitude-longitude grid components （called Yin and Yang respectively） that overlapp each other, and this effectively avoids the coordinate singularity and the grid convergence near the poles. In this overset grid, the way of transferring data between the Yin and Yang components is the key to maintaining the accuracy and robustness in numerical solutions. A numerical interpolation for boundary data exchange, which maintains the accuracy of the original advection scheme and is computationally efficient, is given in this paper. A standard test of the solid-body advection proposed by Williamson is carried out on the Yin-Yang grid. Numerical results show that the quasi-uniform Yin-Yang grid can get around the problems near the poles, and the numerical accuracy in the original semi-Lagrangian scheme is effectively maintained in the Yin-Yang grid.
基金jointly sponsored by the Key Project of the Chinese National Programs for Fundamental Research and Development ("973 Program" Grant No.2013CB430106)+1 种基金the Key Project of the Chinese National Science & Technology Pillar Program during the Twelfth Five-year Plan Period (Grant No.2012BAC22B01)the National Natural Science Foundation of China ( Grant No.41375108)
文摘ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpola tion （referred to as the SL_CL scheme）. The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich＇s semi-Lagrangian advection scheme （referred to as the R2007 scheme）, the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re..R2007 scheme and the SL_CL scheme. The numerical results showed that： （1） the damping effects were remarkably reduced with the Re_R2007 scheme; and （2） the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re..R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with non uniform grid cells, which also verified the Re_R2007 scheme＇s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.
文摘A new non-interpolating semi-Lagrangian scheme has been proposed, which can eliminate any interpolation,and consequently numerical smoothing of forecast fields. Here the new scheme is applied to KdV equation and its performance is assessed by comparing the numerical results with those produced by Ritchie＇s scheme (1986).The comparison shows that the non-interpolating semi-Lagrangian scheme appears to have efficiency advantages.
文摘We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.
基金supported by the National Key Research and Development Program of China (Grant Nos.2017YFC1501901 and 2017YFA0603901)the Beijing Natural Science Foundation (Grant No.JQ18001)。
文摘A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid.Two kinds of moments(i.e.,point values(PV moment) at cell interfaces and volume integrated average(VIA moment) value) are defined within a single cell.The PV moment is updated by a conventional semi-Lagrangian method,while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation.Different from the spatial approximation used in the CSL2(conservative semi-Lagrangian scheme with second order polynomial function) scheme,a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment.To achieve exactly positive-definite preserving,two kinds of corrections are made on the original conservative semi-Lagrangian with rational function(CSLR)scheme.The resulting scheme is inherently conservative,non-negative,and allows a Courant number larger than one.Moreover,the spatial reconstruction can be performed within a single cell,which is very efficient and economical for practical implementation.In addition,a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry,which benefitsthe implementation of the 1 D CSLR solver with large Courant number.The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry.Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical nonnegativity,and it has the potential to transport the tracers accurately in a real atmospheric model.
文摘The interpolation method in a semi-Lagrangian scheme is decisive to its performance. Given the number of grid points one is considering to use for the interpolation, it does not necessarily follow that maximum formal accuracy should give the best results. For the advection equation, the driving force of this method is the method of the characteristics, which accounts for the flow of information in the model equation. This leads naturally to an interpolation problem since the foot point is not in general located on a grid point. We use another interpolation scheme that will allow achieving the high order for the box initial condition.
文摘The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux in time into the integration in space. Compared with the traditional semi-Lagrange scheme, the scheme devised here tries to directly evaluate the average fluxes along cell edges. It is this difference that makes the scheme in this paper simple to implement and easily extend to nonlinear cases. The procedure of evaluation of the average fluxes only depends on the high-order spatial interpolation. Hence the scheme can be implemented as long as the spatial interpolation is available, and no additional temporal discretization is needed. In this paper, the high-order spatial discretization is chosen to be the classical 5th-order weighted essentially non-oscillatory spatial interpolation. In the end, 1D and 2D numerical results show that this method is rather robust. In addition, to exhibit the numerical resolution and efficiency of the proposed scheme, the numerical solutions of the classical 5th-order WENO scheme combined with the 3rd-order Runge-Kutta temporal discretization (WENOJS) are chosen as the reference. We find that the scheme proposed in the paper generates comparable solutions with that of WENOJS, but with less CPU time.
文摘The paper is devised to combine the approximated semi-Lagrange weighted essentially non-oscillatory scheme and flux vector splitting. The approximated finite volume semi-Lagrange that is weighted essentially non-oscillatory scheme with Roe flux had been proposed. The methods using Roe speed to construct the flux probably generates entropy-violating solutions. More seriously, the methods maybe perform numerical instability in two-dimensional cases. A robust and simply remedy is to use a global flux splitting to substitute Roe flux. The combination is tested by several numerical examples. In addition, the comparisons of computing time and resolution between the classical weighted essentially non-oscillatory scheme (WENOJS-LF) and the semi-Lagrange weighted essentially non-oscillatory scheme (WENOEL-LF) which is presented (both combining with the flux vector splitting).
基金funded by the research project STiMulUs,ERC Grant agreement no.278267Financial support has also been provided by the Italian Ministry of Education,University and Research(MIUR)in the frame of the Departments of Excellence Initiative 2018-2022 attributed to DICAM of the University of Trento(Grant L.232/2016)the PRIN2017 project.The authors have also received funding from the University of Trento via the Strategic Initiative Modeling and Simulation.
文摘A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.The staggered DG scheme defines the discrete pressure on the primal triangular mesh,while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh.In this paper,a new pair of equal-order-interpolation velocity-pressure finite elements is proposed.On the primary triangular mesh(the pressure elements),the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle.On the dual mesh instead(the velocity elements),the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries,while they are continuous inside.In other words,the basis functions on the dual mesh arc built by continuous finite elements on the subtriangles.This choice allows the construction of an efficient,quadrature-free and memory saving algorithm.In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations,the arbitrary high order of accuracy in time is achieved through tire use of time-dependent test and basis functions,in combination with simple and efficient Picard iterations.Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes,not only from a computer memory point of view,but also concerning the computational time.
基金Supported by the Ministry of Science and Technology of China (Grant Nos 2006BAC02B01 and 2006BAC03B03)the National High Technology Research and Development Program of China (863 Program) (Grant No 2006AA01A123)
文摘The Global/Regional Assimilation and PrEdiction System(GRAPES) is a newly developed global non-hydrostatic numerical prediction model,which will become the next generation medium-range opera-tional model at China Meteorological Administration(CMA).The dynamic framework of GRAPES is featuring with fully compressible equations,nonhydrostatic or hydrostatic optionally,two-level time semi-Lagrangian and semi-implicit time integration,Charney-Phillips vertical staggering,and complex three-dimensional pre-conditioned Helmholtz solver,etc.Concerning the singularity of horizontal momentum equations at the poles,the polar discretization schemes are described,which include adoption of Arakawa C horizontal grid with ν at poles,incorporation of polar filtering to maintain the computational stability,the correction to Helmholtz equation near the poles,as well as the treatment of semi-Lagrangian interpolation to improve the departure point accuracy,etc.The balanced flow tests validate the rationality of the treatment of semi-Lagrangian departure point calculation and the polar discretization during long time integration.Held and Suarez tests show that the conservation proper-ties of GRAPES model are quite good.
文摘The paper explores the gravity-driven flow of the thin film of a viscoelastic-fluid-based nanofluids(VFBN)along an inclined plane under non-isothermal conditions and subjected to convective cooling at the free-surface.The Newton’s law of cooling is used to model the convective heat-exchange with the ambient at the free-surface.The Giesekus viscoelastic constitutive model,with appropriate modifications to account for non-isothermal effects,is employed to describe the polymeric effects.The unsteady and coupled non-linear partial differential equations(PDEs)describing the model problem are obtained and solved via efficient semi-implicit numerical schemes based on finite difference methods(FDM)implemented in Matlab.The response of the VFBN velocity,temperature,thermal-conductivity and polymeric-stresses to variations in the volume-fraction of embedded nanoparticles is investigated.It is shown that these quantities all increase as the nanoparticle volume-fraction becomes higher.
基金supported by the National Natural Science Foundation of China (Grant No. 41175095)the National Key Technology R&D Program(Grant No. 2012BAC22B01)a research project of the Chinese Academy of Meteorological Sciences (Grant No. 2014Z001)
文摘A 3D dynamic core of the non-hydrostatic model GRAPES（Global/Regional Assimilation and Prediction System） is developed on the Yin-Yang grid to address the polar problem and to enhance the computational efficiency. Three-dimensional Coriolis forcing is introduced to the new core, and full representation of the Coriolis forcing makes it straightforward to share code between the Yin and Yang subdomains. Similar to that in the original GRAPES model, a semi-implicit semi-Lagrangian scheme is adopted for temporal integration and advection with additional arrangement for cross-boundary transport. Under a non-centered second-order temporal and spatial discretization, the dry nonhydrostatic frame is summarized as the solution of an elliptical problem. The resulting Helmholtz equation is solved with the Generalized Conjugate Residual solver in cooperation with the classic Schwarz method. Even though the coefficients of the equation are quite different from those in the original model, the computational procedure of the new core is just the same. The bi-cubic Lagrangian interpolation serves to provide Dirichlet-type boundary conditions with data transfer between the subdomains. The dry core is evaluated with several benchmark test cases, and all the tests display reasonable numerical stability and computing performance. Persistency of the balanced flow and development of both the mountain-induced Rossby wave and Rossby–Haurwitz wave confirms the appropriate installation of the 3D Coriolis terms in the semi-implicit semi-Lagrangian dynamic core on the Yin-Yang grid.
文摘The recent progresses in the research and development of (NWP) in China are reviewed in this paper. The most impressive achievements are the development of direct assimilation of satellite irradiances with a 3DVAR (three-dimentional variational) data assimilation system and a non-hydrostatic modei with a semi-Lagrangian semi-implicit scheme. Progresses have also been made in modei physics and modei application to precipitation and environmental forecasts. Some scientific issues of great importance for further development are discussed.
文摘Quantitative studies of scientific problems require solving correspondent mathematical models. Although a great deal of mathematical models of evolutional problems are set up under continuous space-time meaning, they usually had to be solved numerically after space-time discretization because nonlinear mathematical models except some
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.
