In this paper, a new Riemann-solver-free class of difference schemes are const ructed to 2-D scalar nonlinear hyperbolic conservation laws. We proved thatthese schemes had second order accurate in space and time, and ...In this paper, a new Riemann-solver-free class of difference schemes are const ructed to 2-D scalar nonlinear hyperbolic conservation laws. We proved thatthese schemes had second order accurate in space and time, and satisfied MmB properties under the appropriate CFL limitation. Moreover, these schemes hadbeen extended to systems of 2-D conservation laws. Finally, several numericalexperients show that the performance of these schemes are quite satisfactory.展开更多
In this paper, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation with a variable interfacial parameter, is solved numerically by using a convex splitting scheme which is second-order in time for the non-stoch...In this paper, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation with a variable interfacial parameter, is solved numerically by using a convex splitting scheme which is second-order in time for the non-stochastic part in combination with the Crank- Nicolson and the Adams-Bashforth methods. For the non-stochastic case, the uncondi- tional energy stability is obtained in the sense that a modified energy is non-increasing. The scheme in the stochastic version is then obtained by adding the discretized stochastic term. Numerical experiments are carried out to verify the second-order convergence rate for the non-stochastic case~ and to show the long-time stochastic evolutions using larger time steps.展开更多
In this paper,a fully discrete finite element scheme with second-order temporal accuracy is proposed for a fluid-fluid interaction model,which consists of two Navier-Stokes equations coupled by a linear interface cond...In this paper,a fully discrete finite element scheme with second-order temporal accuracy is proposed for a fluid-fluid interaction model,which consists of two Navier-Stokes equations coupled by a linear interface condition.The proposed fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization,the secondorder backward differentiation formula for temporal discretization,the second-order Gear’s extrapolation approach for the interface terms and extrapolated treatments in linearization for the nonlinear terms.Moreover,the unconditional stability is established by rigorous analysis and error estimate for the fully discrete scheme is also derived.Finally,some numerical experiments are carried out to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.展开更多
在非均匀分层下,目前GRAPES(Global/Regional Assimilation and Prediction System)模式中使用的垂直差分方案只能达到一阶精度。本文设计了一种适用于非均匀分层的二阶精度垂直差分方案,并将它应用于改进GRAPES模式动力框架的垂直离散...在非均匀分层下,目前GRAPES(Global/Regional Assimilation and Prediction System)模式中使用的垂直差分方案只能达到一阶精度。本文设计了一种适用于非均匀分层的二阶精度垂直差分方案,并将它应用于改进GRAPES模式动力框架的垂直离散化过程。一维廓线理想试验结果表明:二阶精度方案可以减少差分计算误差,而这种改进的幅度相对于差分计算本身引起的误差来说仍然是比较小的。通过密度流试验对修改后的模式动力框架进行测试,结果表明二阶方案可以保持模式动力框架的准确性和稳定性。进一步利用实际资料开展批量测试,发现二阶方案可以降低模式高空要素场的预报误差,而且这种改进随着预报时间的延长变得更为明显。最后选择一次典型的华南暴雨过程进行模拟,同样发现二阶精度方案对于48小时之后的降水会有一定程度的改进。展开更多
Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to na...Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at 展开更多
In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the l...In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.展开更多
文摘In this paper, a new Riemann-solver-free class of difference schemes are const ructed to 2-D scalar nonlinear hyperbolic conservation laws. We proved thatthese schemes had second order accurate in space and time, and satisfied MmB properties under the appropriate CFL limitation. Moreover, these schemes hadbeen extended to systems of 2-D conservation laws. Finally, several numericalexperients show that the performance of these schemes are quite satisfactory.
文摘In this paper, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation with a variable interfacial parameter, is solved numerically by using a convex splitting scheme which is second-order in time for the non-stochastic part in combination with the Crank- Nicolson and the Adams-Bashforth methods. For the non-stochastic case, the uncondi- tional energy stability is obtained in the sense that a modified energy is non-increasing. The scheme in the stochastic version is then obtained by adding the discretized stochastic term. Numerical experiments are carried out to verify the second-order convergence rate for the non-stochastic case~ and to show the long-time stochastic evolutions using larger time steps.
基金supported by the Natural Science Foundation of China(grant numbers 11861067 and 11771348)Natural Science Foundation of Xinjiang Province(grant number 2021D01E11).
文摘In this paper,a fully discrete finite element scheme with second-order temporal accuracy is proposed for a fluid-fluid interaction model,which consists of two Navier-Stokes equations coupled by a linear interface condition.The proposed fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization,the secondorder backward differentiation formula for temporal discretization,the second-order Gear’s extrapolation approach for the interface terms and extrapolated treatments in linearization for the nonlinear terms.Moreover,the unconditional stability is established by rigorous analysis and error estimate for the fully discrete scheme is also derived.Finally,some numerical experiments are carried out to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.
文摘在非均匀分层下,目前GRAPES(Global/Regional Assimilation and Prediction System)模式中使用的垂直差分方案只能达到一阶精度。本文设计了一种适用于非均匀分层的二阶精度垂直差分方案,并将它应用于改进GRAPES模式动力框架的垂直离散化过程。一维廓线理想试验结果表明:二阶精度方案可以减少差分计算误差,而这种改进的幅度相对于差分计算本身引起的误差来说仍然是比较小的。通过密度流试验对修改后的模式动力框架进行测试,结果表明二阶方案可以保持模式动力框架的准确性和稳定性。进一步利用实际资料开展批量测试,发现二阶方案可以降低模式高空要素场的预报误差,而且这种改进随着预报时间的延长变得更为明显。最后选择一次典型的华南暴雨过程进行模拟,同样发现二阶精度方案对于48小时之后的降水会有一定程度的改进。
基金P.Li is supported by the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX202711)L.Yang is supported by the Science and Technology Development Fund,Macao SAR(Grant No.0070/2019/A2)+4 种基金the National Natural Science Foundation of China(NSFC)(Grant No.11701598)J.Lan is supported by NSFC(Grant No.11904260)the Natural Science Foundation of Tianjin(Grant No.20JCQNJC02020)R.Du was supported by NSFC(Grant No.11501399)J.Chen is supported by NSFC(Grant No.11971021).
文摘Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at
基金This work is supported in part by the grants NSFC 12071090(W.Chen)NSF DMS-2012669(C.Wang)+2 种基金NSFC 11871159Guangdong Provincial Key Laboratory for Computational Science and Material Design 2019B030301001(X.Wang)NSF DMS-1719854,DMS-2012634(S.Wise).C.Wang also thanks the Key Laboratory of Mathematics for Nonlinear Sciences,Fudan University,for the support.
文摘In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.
