We construct new fifth-order alternative WENO(A-WENO)schemes for the Euler equations of gas dynamics.The new scheme is based on a new adaptive diffusion centralupwind Rankine-Hugoniot(CURH)numerical flux.The CURH nume...We construct new fifth-order alternative WENO(A-WENO)schemes for the Euler equations of gas dynamics.The new scheme is based on a new adaptive diffusion centralupwind Rankine-Hugoniot(CURH)numerical flux.The CURH numerical fluxes have been recently proposed in[Garg et al.J Comput Phys 428,2021]in the context of secondorder semi-discrete finite-volume methods.The proposed adaptive diffusion CURH flux contains a smaller amount of numerical dissipation compared with the adaptive diffusion central numerical flux,which was also developed with the help of the discrete RankineHugoniot conditions and used in the fifth-order A-WENO scheme recently introduced in[Wang et al.SIAM J Sci Comput 42,2020].As in that work,we here use the fifth-order characteristic-wise WENO-Z interpolations to evaluate the fifth-order point values required by the numerical fluxes.The resulting one-and two-dimensional schemes are tested on a number of numerical examples,which clearly demonstrate that the new schemes outperform the existing fifth-order A-WENO schemes without compromising the robustness.展开更多
We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by sol...We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh.When the evolution step is complete,the grid points are redistributed according to the moving mesh differential equation.Finally,the evolved solution is projected onto the new mesh in a conservative manner.The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems.Our numerical results demonstrate that in both cases,the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.展开更多
Non-equilibrium hyperbolic traffic models can be derived as continuum approximations of car-following models and in many cases the resulting continuum models are non-conservative.This leads to numerical difficulties,w...Non-equilibrium hyperbolic traffic models can be derived as continuum approximations of car-following models and in many cases the resulting continuum models are non-conservative.This leads to numerical difficulties,which seem to have discouraged further development of complex behavioral continuum models,which is a significant research need.In this paper,we develop a robust numerical scheme that solves hyperbolic traffic flow models based on their non-conservative form.We develop a fifth-order alternative weighted essentially non-oscillatory(A-WENO)finite-difference scheme based on the path-conservative central-upwind(PCCU)method for several non-equilibrium traffic flow models.In order to treat the non-conservative product terms,we use a path-conservative technique.To this end,we first apply the recently proposed secondorder finite-volume PCCU scheme to the traffic flow models,and then extend this scheme to the fifth-order of accuracy via the finite-difference A-WENO framework.The designed schemes are applied to three different traffic flow models and tested on a number of challenging numerical examples.Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher.The proposed scheme in this paper sets the stage for developing more robust and complex continuum traffic flow models with respect to human psychological factors.展开更多
We present an extension of the flux globalization based well-balanced pathconservative central-upwind scheme to the one-and two-dimensional thermal rotating shallow water equations.The scheme is well-balanced in the s...We present an extension of the flux globalization based well-balanced pathconservative central-upwind scheme to the one-and two-dimensional thermal rotating shallow water equations.The scheme is well-balanced in the sense that it can exactly preserve a variety of physically relevant steady states.In the one-dimensional case,it can preserve different“lake-at-rest”equilibria,thermo-geostrophic equilibria,as well as general moving-water steady states.In the two-dimensional case,preserving general moving-water steady states is difficult,and to the best of our knowledge,none of existing schemes can achieve this ultimate goal.The proposed scheme can exactly preserve the x-and y-directional jets in the rotational frame as well as certain genuinely two-dimensional equilibria.Furthermore,our approach employs a path-conservative technique for discretizing nonconservative product terms,which are incorporated into the global fluxes.This allows the developed scheme to exactly preserve some of the discontinuous steady states as well.We provide a number of numerical examples to demonstrate the advantages of the proposed scheme over some alternative finitevolume methods.展开更多
We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution...We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution,simplicity,universality and robustness.At the same time,the central-upwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes.In this paper,we present a modification of the one-dimensional fully-and semi-discrete central-upwind schemes,in which the numerical dissipation is reduced even further.The goal is achieved by a more accurate projection of the evolved quantities onto the original grid.In the semi-discrete case,the reduction of dissipation procedure leads to a new,less dissipative numerical flux.We also extend the new semi-discrete scheme to the twodimensional case via the rigorous,genuinely multidimensional derivation.The new semi-discrete schemes are tested on a number of numerical examples,where one can observe an improved resolution,especially of the contact waves.展开更多
This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expec...This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expected.The problem arises from over-restriction of some slope limiters,which keep slopes between interfaces of cells to be Total-Variation-Diminishing.This study reports the defect and presents a re-derived optimal formula.Numerical experiments highlight the significance of this formula,especially in long-time,large-scale simulations.展开更多
给出了求解一维双曲型守恒律的一种半离散三阶中心迎风格式,并利用逐维进行计算的方法将格式推广到二维守恒律。构造格式时利用了波传播的单侧局部速度,三阶重构方法的引入保证了格式的精度。时间方向的离散采用三阶TVD R unge-K u tta...给出了求解一维双曲型守恒律的一种半离散三阶中心迎风格式,并利用逐维进行计算的方法将格式推广到二维守恒律。构造格式时利用了波传播的单侧局部速度,三阶重构方法的引入保证了格式的精度。时间方向的离散采用三阶TVD R unge-K u tta方法。本文格式保持了中心差分格式简单的优点,即不需用R iem ann解算器,避免了进行特征分解过程。用该格式对一维和二维守恒律进行了大量的数值试验,结果表明本文格式是高精度、高分辨率的。展开更多
基金The work of B.S.Wang and W.S.Don was partially supported by the Ocean University of China through grant 201712011The work of A.Kurganov was supported in part by NSFC grants 11771201 and 1201101343by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘We construct new fifth-order alternative WENO(A-WENO)schemes for the Euler equations of gas dynamics.The new scheme is based on a new adaptive diffusion centralupwind Rankine-Hugoniot(CURH)numerical flux.The CURH numerical fluxes have been recently proposed in[Garg et al.J Comput Phys 428,2021]in the context of secondorder semi-discrete finite-volume methods.The proposed adaptive diffusion CURH flux contains a smaller amount of numerical dissipation compared with the adaptive diffusion central numerical flux,which was also developed with the help of the discrete RankineHugoniot conditions and used in the fifth-order A-WENO scheme recently introduced in[Wang et al.SIAM J Sci Comput 42,2020].As in that work,we here use the fifth-order characteristic-wise WENO-Z interpolations to evaluate the fifth-order point values required by the numerical fluxes.The resulting one-and two-dimensional schemes are tested on a number of numerical examples,which clearly demonstrate that the new schemes outperform the existing fifth-order A-WENO schemes without compromising the robustness.
基金The work of A.Kurganov was supported in part by the National Natural Science Foundation of China grant 11771201by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh.When the evolution step is complete,the grid points are redistributed according to the moving mesh differential equation.Finally,the evolved solution is projected onto the new mesh in a conservative manner.The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems.Our numerical results demonstrate that in both cases,the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.
基金NSFC grants 12171226 and 12111530004the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘Non-equilibrium hyperbolic traffic models can be derived as continuum approximations of car-following models and in many cases the resulting continuum models are non-conservative.This leads to numerical difficulties,which seem to have discouraged further development of complex behavioral continuum models,which is a significant research need.In this paper,we develop a robust numerical scheme that solves hyperbolic traffic flow models based on their non-conservative form.We develop a fifth-order alternative weighted essentially non-oscillatory(A-WENO)finite-difference scheme based on the path-conservative central-upwind(PCCU)method for several non-equilibrium traffic flow models.In order to treat the non-conservative product terms,we use a path-conservative technique.To this end,we first apply the recently proposed secondorder finite-volume PCCU scheme to the traffic flow models,and then extend this scheme to the fifth-order of accuracy via the finite-difference A-WENO framework.The designed schemes are applied to three different traffic flow models and tested on a number of challenging numerical examples.Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher.The proposed scheme in this paper sets the stage for developing more robust and complex continuum traffic flow models with respect to human psychological factors.
基金supported in part by China Postdoctoral Science Foundation(No.2022M721481)The work of A.Kurganov was supported in part by NSFC grant 12171226 and by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001)The work of Y.Liu was supported in part by SNFS grants 200020204917 and FZEB-0-166980.
文摘We present an extension of the flux globalization based well-balanced pathconservative central-upwind scheme to the one-and two-dimensional thermal rotating shallow water equations.The scheme is well-balanced in the sense that it can exactly preserve a variety of physically relevant steady states.In the one-dimensional case,it can preserve different“lake-at-rest”equilibria,thermo-geostrophic equilibria,as well as general moving-water steady states.In the two-dimensional case,preserving general moving-water steady states is difficult,and to the best of our knowledge,none of existing schemes can achieve this ultimate goal.The proposed scheme can exactly preserve the x-and y-directional jets in the rotational frame as well as certain genuinely two-dimensional equilibria.Furthermore,our approach employs a path-conservative technique for discretizing nonconservative product terms,which are incorporated into the global fluxes.This allows the developed scheme to exactly preserve some of the discontinuous steady states as well.We provide a number of numerical examples to demonstrate the advantages of the proposed scheme over some alternative finitevolume methods.
基金supported in part by the NSF Grant DMS-0310585The work of C.-T.Lin was supported in part by the NSC grants NSC 94-2115-M-126-003 and 91-2115-M-126-001.
文摘We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution,simplicity,universality and robustness.At the same time,the central-upwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes.In this paper,we present a modification of the one-dimensional fully-and semi-discrete central-upwind schemes,in which the numerical dissipation is reduced even further.The goal is achieved by a more accurate projection of the evolved quantities onto the original grid.In the semi-discrete case,the reduction of dissipation procedure leads to a new,less dissipative numerical flux.We also extend the new semi-discrete scheme to the twodimensional case via the rigorous,genuinely multidimensional derivation.The new semi-discrete schemes are tested on a number of numerical examples,where one can observe an improved resolution,especially of the contact waves.
文摘This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expected.The problem arises from over-restriction of some slope limiters,which keep slopes between interfaces of cells to be Total-Variation-Diminishing.This study reports the defect and presents a re-derived optimal formula.Numerical experiments highlight the significance of this formula,especially in long-time,large-scale simulations.
文摘给出了求解一维双曲型守恒律的一种半离散三阶中心迎风格式,并利用逐维进行计算的方法将格式推广到二维守恒律。构造格式时利用了波传播的单侧局部速度,三阶重构方法的引入保证了格式的精度。时间方向的离散采用三阶TVD R unge-K u tta方法。本文格式保持了中心差分格式简单的优点,即不需用R iem ann解算器,避免了进行特征分解过程。用该格式对一维和二维守恒律进行了大量的数值试验,结果表明本文格式是高精度、高分辨率的。
