In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the g...In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.展开更多
We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxatio...We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations. To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method.展开更多
The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,wh...The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.展开更多
We study the asymptotic-preserving fully discrete schemes for nonequilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry.The research is based on two-temperature models with Larsen’s f...We study the asymptotic-preserving fully discrete schemes for nonequilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry.The research is based on two-temperature models with Larsen’s flux-limited diffusion operators.Finite volume spatially discrete schemes are developed to circumvent the singularity at the origin and the polar axis and assure local conservation.Asymmetric second order accurate spatial approximation is utilized instead of the traditional first order one for boundary flux-limiters to consummate the schemes with higher order global consistency errors.The harmonic average approach in spherical geometry is analyzed,and its second order accuracy is demonstrated.By formal analysis,we prove these schemes and their corresponding fully discrete schemes with implicitly balanced and linearly implicit time evolutions have first order asymptoticpreserving properties.By designing associated manufactured solutions and reference solutions,we verify the desired performance of the fully discrete schemes with numerical tests,which illustrates quantitatively they are first order asymptotic-preserving and basically second order accurate,hence competent for simulations of both equilibrium and non-equilibrium radiation diffusion problems.展开更多
We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau(FPL)equation that works uniformly from kinetic to fluid regimes.Such a scheme inevitably needs an implicit discretization of the nonlinea...We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau(FPL)equation that works uniformly from kinetic to fluid regimes.Such a scheme inevitably needs an implicit discretization of the nonlinear collision operator,which is difficult to invert.Inspired by work[9]we seek a linear operator to penalize the quantum FPL collision term QqFPL in order to remove the stiffness induced by the small Knudsen number.However,there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians(Bose-Einstein or Fermi-Dirac distribution).In this paper,we propose to penalize QqFPL by the”classical”linear Fokker-Planck operator.It is based on the observation that the classicalMaxwellian,with the temperature replaced by the internal energy,has the same first five moments as the quantum Maxwellian.Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both fluid and kinetic regimes.展开更多
This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem.The numerical method is designed for arbitrary spacedependent anisotropy directions and does not require any specially ...This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem.The numerical method is designed for arbitrary spacedependent anisotropy directions and does not require any specially adapted coordinate system.It is also designed to be equally accurate in the strongly and the mildly anisotropic cases.The method is applied to the Euler-Lorentz system,in the drift-fluid limit.This system provides a model for magnetized plasmas.展开更多
A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms...A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes （meshes that do not resolve the mean free path） can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.展开更多
辐射输运方程的数值模拟在天体物理、武器物理和惯性约束与磁约束聚变等研究中都起着非常重要的作用.在实际问题中,背景介质的不透明度系数决定了辐射光子在其中的传输行为.光性薄(不透明度系数小)的介质对辐射光子是透明的,光子与背景...辐射输运方程的数值模拟在天体物理、武器物理和惯性约束与磁约束聚变等研究中都起着非常重要的作用.在实际问题中,背景介质的不透明度系数决定了辐射光子在其中的传输行为.光性薄(不透明度系数小)的介质对辐射光子是透明的,光子与背景介质的相互作用弱,光子传输具有输运传播性质;而光性厚(不透明度系数大)的介质对辐射光子是不透明的,光子与背景介质的相互作用强,光子传输具有扩散性质.因此在辐射输运方程的计算中,如何设计既能得到光子输运传播性质又能捕捉光子扩散传播性质的渐近保持离散格式是目前一个非常活跃和前沿的研究方向.本文简要介绍近几年在辐射输运方程的渐近保持统一气体动理学格式(unified gas kinetic scheme,UGKS)研究方面的进展.本文主要以灰体辐射输运方程为例,详细介绍UGKS的构造方法并给出其渐近分析.同时,结合角度有限元方法和球谐函数展开的方法,介绍如何减弱/去除基于离散纵标法的UGKS具有射线效应的问题,以及相应的改进渐近保持格式.此外,也介绍了将渐近保持的UGKS应用拓展到考虑流体运动的完全辐射流体力学方程组.最后,用一些数值例子验证了格式的渐近保持性和保正性等性质.展开更多
In this paper,we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes.Different from the former scheme [J.Comput.Phys....In this paper,we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes.Different from the former scheme [J.Comput.Phys.285（2015）,265-279] on uniform meshes,in this paper,in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme（UGKS）,we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations.We can prove that the scheme is asymptotic preserving,and especially for the distorted quadrilateral meshes,a nine-point scheme [SIAM J.SCI.COMPUT.30（2008）,1341-1361] for the diffusion limit equations is obtained,which is naturally reduced to standard five-point scheme for the orthogonal meshes.The numerical examples on distorted meshes are included to validate the current approach.展开更多
基金supported by the Simons Foundation:Collaboration Grantssupported by the AFOSR grant FA9550-18-1-0383.
文摘In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.
基金This work was supported by the German research foundation DFG under grant KL 1105/9-1.
文摘We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations. To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method.
基金the National Numerical Wind Tunnel Project(No.NNW2018-ZT4A08)the National Natural Science Foundation of China(Nos.11872323 and 11971025)the Natural Science Foundation of Fujian Province(No.2019J06002)。
文摘The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.
基金The authors are very grateful to the editors and the anonymous referees for helpful suggestions to enhance the paper.This work is supported by the National Natural Science Foundation of China(11271054,11471048,11571048,U1630249)the Science Foundation of CAEP(2014A0202010)the Science Challenge Project(No.JCKY2016212A502)and the Foundation of LCP.
文摘We study the asymptotic-preserving fully discrete schemes for nonequilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry.The research is based on two-temperature models with Larsen’s flux-limited diffusion operators.Finite volume spatially discrete schemes are developed to circumvent the singularity at the origin and the polar axis and assure local conservation.Asymmetric second order accurate spatial approximation is utilized instead of the traditional first order one for boundary flux-limiters to consummate the schemes with higher order global consistency errors.The harmonic average approach in spherical geometry is analyzed,and its second order accuracy is demonstrated.By formal analysis,we prove these schemes and their corresponding fully discrete schemes with implicitly balanced and linearly implicit time evolutions have first order asymptoticpreserving properties.By designing associated manufactured solutions and reference solutions,we verify the desired performance of the fully discrete schemes with numerical tests,which illustrates quantitatively they are first order asymptotic-preserving and basically second order accurate,hence competent for simulations of both equilibrium and non-equilibrium radiation diffusion problems.
基金supported by NSF grant DMS-0608720 and NSF FRG grant DMS-0757285.S.
文摘We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau(FPL)equation that works uniformly from kinetic to fluid regimes.Such a scheme inevitably needs an implicit discretization of the nonlinear collision operator,which is difficult to invert.Inspired by work[9]we seek a linear operator to penalize the quantum FPL collision term QqFPL in order to remove the stiffness induced by the small Knudsen number.However,there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians(Bose-Einstein or Fermi-Dirac distribution).In this paper,we propose to penalize QqFPL by the”classical”linear Fokker-Planck operator.It is based on the observation that the classicalMaxwellian,with the temperature replaced by the internal energy,has the same first five moments as the quantum Maxwellian.Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both fluid and kinetic regimes.
基金supported by the Marie Curie Actions of the EuropeanCommission in the frame of the DEASE project(MEST-CT-2005-021122)by the”F´ed´eration de recherche CNRS sur la fusion par confinementmagn´etique”,by theAssociation Euratom-CEA in the framework of the contract”Gyro-AP”(contract#V3629.001 avenant 1)by the University Paul Sabatier in the frame of the contract”MOSITER”.This work was performed while the first author held a post-doctoral position funded by the Fondation”Sciences et Technologies pour l’A´eronautique et l’Espace”,in the frame of the project”Plasmax”(contract#RTRA-STAE/2007/PF/002).
文摘This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem.The numerical method is designed for arbitrary spacedependent anisotropy directions and does not require any specially adapted coordinate system.It is also designed to be equally accurate in the strongly and the mildly anisotropic cases.The method is applied to the Euler-Lorentz system,in the drift-fluid limit.This system provides a model for magnetized plasmas.
文摘A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes （meshes that do not resolve the mean free path） can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.
文摘辐射输运方程的数值模拟在天体物理、武器物理和惯性约束与磁约束聚变等研究中都起着非常重要的作用.在实际问题中,背景介质的不透明度系数决定了辐射光子在其中的传输行为.光性薄(不透明度系数小)的介质对辐射光子是透明的,光子与背景介质的相互作用弱,光子传输具有输运传播性质;而光性厚(不透明度系数大)的介质对辐射光子是不透明的,光子与背景介质的相互作用强,光子传输具有扩散性质.因此在辐射输运方程的计算中,如何设计既能得到光子输运传播性质又能捕捉光子扩散传播性质的渐近保持离散格式是目前一个非常活跃和前沿的研究方向.本文简要介绍近几年在辐射输运方程的渐近保持统一气体动理学格式(unified gas kinetic scheme,UGKS)研究方面的进展.本文主要以灰体辐射输运方程为例,详细介绍UGKS的构造方法并给出其渐近分析.同时,结合角度有限元方法和球谐函数展开的方法,介绍如何减弱/去除基于离散纵标法的UGKS具有射线效应的问题,以及相应的改进渐近保持格式.此外,也介绍了将渐近保持的UGKS应用拓展到考虑流体运动的完全辐射流体力学方程组.最后,用一些数值例子验证了格式的渐近保持性和保正性等性质.
基金supported by the Science and Technology Development foundation of China Academy of Engineering Physics(Grant Nos.2015B0202041,2015B0202040)the Science and Technology Development foundation of China Academy of Engineering Physics(Grant 2015B0202040)+2 种基金the Science and Technology Development foundation of China Academy of Engineering Physics(Grant No.2015B0202033)for LiNSFC(Grant No.11371068)for SunNSFC(Grant No.11371068)for Zeng
文摘In this paper,we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes.Different from the former scheme [J.Comput.Phys.285（2015）,265-279] on uniform meshes,in this paper,in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme（UGKS）,we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations.We can prove that the scheme is asymptotic preserving,and especially for the distorted quadrilateral meshes,a nine-point scheme [SIAM J.SCI.COMPUT.30（2008）,1341-1361] for the diffusion limit equations is obtained,which is naturally reduced to standard five-point scheme for the orthogonal meshes.The numerical examples on distorted meshes are included to validate the current approach.
