Orbital-free density functional theory(OFDFT)is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions.We examine some popular algorithms for optimizing ...Orbital-free density functional theory(OFDFT)is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions.We examine some popular algorithms for optimizing the electron density distribution in OFDFT,explaining their suitability,benchmarking their performance,and suggesting some improvements.We start by describing the constrained optimization problem that encompasses electron density optimization.Next,we discuss the line search(including Wolfe conditions)and the nonlinear conjugate gradient and truncated Newton algorithms,as implemented in our open source OFDFT code.We finally focus on preconditioners derived from OFDFT energy functionals.Newlyderived preconditioners are successful for simulation cells of all sizes without regions of low electron-density and for small simulation cells with such regions.展开更多
We investigate the validity of stationary simulations for semiconductor quantum charge transport in a one-dimensional resonant tunneling diode via fluid type models.Careful numerical investigations to a quantum hydrod...We investigate the validity of stationary simulations for semiconductor quantum charge transport in a one-dimensional resonant tunneling diode via fluid type models.Careful numerical investigations to a quantum hydrodynamic model reveal that the transient simulations do not always converge to the steady states.In particular,growing oscillations are observed at relatively large applied voltage.A dynamical bifurcation is responsible for the stability interchange of the steady state.Transient and stationary computations are also performed for a unipolar quantum drift-diffusion model.展开更多
The iterative solution of the sequence of linear systems arising from threetemperature(3-T)energy equations is an essential component in the numerical simulation of radiative hydrodynamic(RHD)problem.However,due to th...The iterative solution of the sequence of linear systems arising from threetemperature(3-T)energy equations is an essential component in the numerical simulation of radiative hydrodynamic(RHD)problem.However,due to the complicated application features of the RHD problems,solving 3-T linear systems with classical preconditioned iterative techniques is challenging.To address this difficulty,a physicalvariable based coarsening two-level(PCTL)preconditioner has been proposed by dividing the fully coupled system into four individual easier-to-solve subsystems.Despite its nearly optimal complexity and robustness,the PCTL algorithm suffers from poor efficiency because of the overhead associatedwith the construction of setup phase and the solution of subsystems.Furthermore,the PCTL algorithm employs a fixed strategy for solving the sequence of 3-T linear systems,which completely ignores the dynamically and slowly changing features of these linear systems.To address these problems and to efficiently solve the sequence of 3-T linear systems,we propose an adaptive two-level preconditioner based on the PCTL algorithm,referred to as αSetup-PCTL.The adaptive strategies of the αSetup-PCTL algorithm are inspired by those of αSetup-AMG algorithm,which is an adaptive-setup-based AMG solver for sequence of sparse linear systems.The proposed αSetup-PCTL algorithm could adaptively employ the appropriate strategies for each linear system,and thus increase the overall efficiency.Numerical results demonstrate that,for 36 linear systems,the αSetup-PCTL algorithm achieves an average speedup of 2.2,and a maximum speedup of 4.2 when compared to the PCTL algorithm.展开更多
Based on the theory of optimization,we use edges and angles of cells to represent the geometric quality of computational grids,employ the local gradients of the flow variables to describe the variation of flow field,a...Based on the theory of optimization,we use edges and angles of cells to represent the geometric quality of computational grids,employ the local gradients of the flow variables to describe the variation of flow field,and construct a multi-objective programming model.The solution of this optimization problem gives appropriate balance between the geometric quality and adaptation of grids.By solving the optimization problem,we propose a new grid rezoning method,which not only keeps good geometric quality of grids,but also can track rapid changes in the flow field.In particular,it performs well for some complex concave domains with corners.We also incorporate the rezoningmethod into anArbitrary Lagrangian-Eulerian(ALE)method which is widely used in the simulation of high-speed multi-material flows.The proposed rezoning and ALE methods of this paper are tested by a number of numerical examples with complex concave domains and compared with some other rezoning methods.The numerical results validate the robustness of the proposed methods.展开更多
We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unstea...We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.展开更多
In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation la...In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.展开更多
The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entro...The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.展开更多
In this paper,a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species BGK model in[A.D.Kotelnikov and D.C.Montgomery,A Kinetic Method for Computing Inhomogeneous Fluid ...In this paper,a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species BGK model in[A.D.Kotelnikov and D.C.Montgomery,A Kinetic Method for Computing Inhomogeneous Fluid Behavior,J.Comput.Phys.134(1997)364-388].Different from the conventional BGK model,the collisions between different species are taken into consideration.Based on the Chapman-Enskog expansion,the corresponding macroscopic equations are derived from this two-species model.Because of the relaxation terms in the governing equations,the method of operator splitting is applied.In the hyperbolic part,the integral solutions of the BGK equations are used to construct the numerical fluxes at the cell interface in the framework of finite volume method.Numerical tests are presented in this paper to validate the current approach for the compressible multicomponent flows.The theoretical analysis on the spurious oscillations at the interface is also presented.展开更多
We construct a nonlinear monotone finite volume scheme for threedimensional diffusion equation on tetrahedral meshes.Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme,we ...We construct a nonlinear monotone finite volume scheme for threedimensional diffusion equation on tetrahedral meshes.Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme,we present a new efficient eliminating method.The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously.The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes,and also show that our scheme appears to be approximate second-order accuracy for solution.展开更多
Quantum molecular dynamic simulations have been employed to study the equation of state(EOS)of fluid helium under shock compressions.The principal Hugoniot is determined from EOS,where corrections from atomic ionizati...Quantum molecular dynamic simulations have been employed to study the equation of state(EOS)of fluid helium under shock compressions.The principal Hugoniot is determined from EOS,where corrections from atomic ionization are added onto the calculated data.Our simulation results indicate that principal Hugoniot shows good agreementwith gas gun and laser driven experiments,andmaximum compression ratio of 5.16 is reached at 106 GPa.展开更多
Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh.There are three weights including the arithmetic,the ...Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh.There are three weights including the arithmetic,the harmonic,and the geometric weight in the weighted discontinuous Galerkin scheme.For the time discretization,we treat the nonlinear diffusion coefficients explicitly,and apply the semiimplicit integration factormethod to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization.The semi-implicit integration factor method can not only avoid severe timestep limits,but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method.Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.展开更多
The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relationwhich often involvesmany independent variables.The constitutive relation of a polymeric fluid is a function of six...The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relationwhich often involvesmany independent variables.The constitutive relation of a polymeric fluid is a function of six variables,even after making the simplifying assumption that stress depends only on the rate of strain.Precomputing such a function is usually considered too expensive.Consequently the value of sequential multiscale modeling is often limited to“parameter passing”.Here we demonstrate that sparse representations can be used to drastically reduce the computational cost for precomputing functions of many variables.This strategy dramatically increases the efficiency of sequential multiscale modeling,making it very competitive in many situations.展开更多
In this paper,a direct arbitrary Lagrangian-Eulerian(ALE)discontinuous Galerkin(DG)scheme is proposed for simulating compressible multi-material flows on the adaptive quadrilateral meshes.Our scheme couples a conserva...In this paper,a direct arbitrary Lagrangian-Eulerian(ALE)discontinuous Galerkin(DG)scheme is proposed for simulating compressible multi-material flows on the adaptive quadrilateral meshes.Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture.The coupled system is discretized in the reference element and we use a kind of Taylor expansion basis functions to construct the interpolation polynomials of the variables.We show the property that the material derivatives of the basis functions in the DG discretization are equal to zero,with which the scheme is simplified.In addition,the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from[H.Z.Tang and T.Tang,Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws,SIAMJ.NUMER.ANAL].This adaptivemesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation near the material interfaces in the simulations.With the help of this adaptivemesh method,the resolution of the solution near the target regions can be greatly improved and the computational efficiency of the simulation is increased.Our scheme can be applied in the simulations for the gas andwatermedia efficiently,and it ismore concise compared to some other methods such as the indirect ALE methods.Several examples including the gas-water flow problemare presented to demonstrate the efficiency of our scheme,and the results show that our scheme can capture the wave structures sharply with high robustness.展开更多
基金We would like to thank the National Defense Science and Engineering Graduate Fellowship program(L.H.)and the National Science Foundation(E.A.C.)for funding.
文摘Orbital-free density functional theory(OFDFT)is a quantum mechanical method in which the energy of a material depends only on the electron density and ionic positions.We examine some popular algorithms for optimizing the electron density distribution in OFDFT,explaining their suitability,benchmarking their performance,and suggesting some improvements.We start by describing the constrained optimization problem that encompasses electron density optimization.Next,we discuss the line search(including Wolfe conditions)and the nonlinear conjugate gradient and truncated Newton algorithms,as implemented in our open source OFDFT code.We finally focus on preconditioners derived from OFDFT energy functionals.Newlyderived preconditioners are successful for simulation cells of all sizes without regions of low electron-density and for small simulation cells with such regions.
基金This research is partially supported by NSFC under grant No.90407021National Basic Research Pro-gram of China under contract number 2007CB814800the China Ministry of Educa-tion under contract number NCET-06-0011.
文摘We investigate the validity of stationary simulations for semiconductor quantum charge transport in a one-dimensional resonant tunneling diode via fluid type models.Careful numerical investigations to a quantum hydrodynamic model reveal that the transient simulations do not always converge to the steady states.In particular,growing oscillations are observed at relatively large applied voltage.A dynamical bifurcation is responsible for the stability interchange of the steady state.Transient and stationary computations are also performed for a unipolar quantum drift-diffusion model.
基金financially supported by the National Natural Science Foundation of China(62032023 and 11971414)Hunan National Applied Mathematics Center(2020ZYT003)the Research Foundation of Education Bureau of Hunan(21B0162).
文摘The iterative solution of the sequence of linear systems arising from threetemperature(3-T)energy equations is an essential component in the numerical simulation of radiative hydrodynamic(RHD)problem.However,due to the complicated application features of the RHD problems,solving 3-T linear systems with classical preconditioned iterative techniques is challenging.To address this difficulty,a physicalvariable based coarsening two-level(PCTL)preconditioner has been proposed by dividing the fully coupled system into four individual easier-to-solve subsystems.Despite its nearly optimal complexity and robustness,the PCTL algorithm suffers from poor efficiency because of the overhead associatedwith the construction of setup phase and the solution of subsystems.Furthermore,the PCTL algorithm employs a fixed strategy for solving the sequence of 3-T linear systems,which completely ignores the dynamically and slowly changing features of these linear systems.To address these problems and to efficiently solve the sequence of 3-T linear systems,we propose an adaptive two-level preconditioner based on the PCTL algorithm,referred to as αSetup-PCTL.The adaptive strategies of the αSetup-PCTL algorithm are inspired by those of αSetup-AMG algorithm,which is an adaptive-setup-based AMG solver for sequence of sparse linear systems.The proposed αSetup-PCTL algorithm could adaptively employ the appropriate strategies for each linear system,and thus increase the overall efficiency.Numerical results demonstrate that,for 36 linear systems,the αSetup-PCTL algorithm achieves an average speedup of 2.2,and a maximum speedup of 4.2 when compared to the PCTL algorithm.
文摘Based on the theory of optimization,we use edges and angles of cells to represent the geometric quality of computational grids,employ the local gradients of the flow variables to describe the variation of flow field,and construct a multi-objective programming model.The solution of this optimization problem gives appropriate balance between the geometric quality and adaptation of grids.By solving the optimization problem,we propose a new grid rezoning method,which not only keeps good geometric quality of grids,but also can track rapid changes in the flow field.In particular,it performs well for some complex concave domains with corners.We also incorporate the rezoningmethod into anArbitrary Lagrangian-Eulerian(ALE)method which is widely used in the simulation of high-speed multi-material flows.The proposed rezoning and ALE methods of this paper are tested by a number of numerical examples with complex concave domains and compared with some other rezoning methods.The numerical results validate the robustness of the proposed methods.
基金financially supported by Hunan National Applied Mathematics Center(2020ZYT003)National Natural Science Foundation of China(11971414,62102167)+1 种基金Research Foundation of Education Bureau of Hunan(21B0162)Guangdong Basic and Applied Basic Research Foundation(2020A1515110364).
文摘We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.
基金This work was supported by the National Natural Science Foundation of China(Nos.91530118,91130003,11021101,11290142,11471310,11601032,11301234,11271171)the Provincial Natural Science Foundation of Jiangxi(Nos.20142BCB23009,20161ACB20006,20151BAB201012).
文摘In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.
基金Nation Key R&D Program of China(Grant No.2022YFA1004500)and NSFC(Grant No.12072043).
文摘The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.
基金Natural Science Foundation of China(NSFC)No.10931004,No.11171037 and No.91130021.
文摘In this paper,a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species BGK model in[A.D.Kotelnikov and D.C.Montgomery,A Kinetic Method for Computing Inhomogeneous Fluid Behavior,J.Comput.Phys.134(1997)364-388].Different from the conventional BGK model,the collisions between different species are taken into consideration.Based on the Chapman-Enskog expansion,the corresponding macroscopic equations are derived from this two-species model.Because of the relaxation terms in the governing equations,the method of operator splitting is applied.In the hyperbolic part,the integral solutions of the BGK equations are used to construct the numerical fluxes at the cell interface in the framework of finite volume method.Numerical tests are presented in this paper to validate the current approach for the compressible multicomponent flows.The theoretical analysis on the spurious oscillations at the interface is also presented.
基金The authors thank two reviewers for their numerous constructive comments and suggestions that helped to improve the paper significantly.This work is partially supported by NSAF(No.U1430101)the National Natural Science Foundation of China(91330106,11571047,11571048,11401034)+2 种基金China Postdoctoral Science Foundation(20110490328)the natural science foundation of Shandong Province(ZR2012AM019,ZR2013AM023,ZR2014AM013)Independent innovation foundation of Shandong University(2012TS018).
文摘We construct a nonlinear monotone finite volume scheme for threedimensional diffusion equation on tetrahedral meshes.Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme,we present a new efficient eliminating method.The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously.The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes,and also show that our scheme appears to be approximate second-order accuracy for solution.
基金supported by NSFC under Grant No.11005012,by the National Basic Security Research Program of China,and by the National High-Tech ICF Committee of China.
文摘Quantum molecular dynamic simulations have been employed to study the equation of state(EOS)of fluid helium under shock compressions.The principal Hugoniot is determined from EOS,where corrections from atomic ionization are added onto the calculated data.Our simulation results indicate that principal Hugoniot shows good agreementwith gas gun and laser driven experiments,andmaximum compression ratio of 5.16 is reached at 106 GPa.
基金the National Nature Science Foundation of China(11171038)R.Zhang’s work was also supported by Brazilian Young Talent Attraction Program via National Council for Scientific and Technological Development(CNPq).J.Zhu and A.Loula’s works were partially supported by CNPq.X.Cui’s work was partially supported by the National Natural Science Foundation of China(11271054)+1 种基金the Science Foundation of CAEP(2010A0202010,2012B0202026)the Defense Industrial Technology Development Program(B1520110011).
文摘Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh.There are three weights including the arithmetic,the harmonic,and the geometric weight in the weighted discontinuous Galerkin scheme.For the time discretization,we treat the nonlinear diffusion coefficients explicitly,and apply the semiimplicit integration factormethod to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization.The semi-implicit integration factor method can not only avoid severe timestep limits,but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method.Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.
基金The work of Carlos J.Garcıa-Cervera is supported in part by NSF grants DMS-0411504 and DMS-0505738The work of Weiqing Ren is supported in part by NSF grant DMS-0604382The work of Jianfeng Lu and Weinan E is supported in part by ONR grant N00014-01-0674,DOE grant DE-FG02-03ER25587 and NSF grant DMS-0407866.
文摘The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relationwhich often involvesmany independent variables.The constitutive relation of a polymeric fluid is a function of six variables,even after making the simplifying assumption that stress depends only on the rate of strain.Precomputing such a function is usually considered too expensive.Consequently the value of sequential multiscale modeling is often limited to“parameter passing”.Here we demonstrate that sparse representations can be used to drastically reduce the computational cost for precomputing functions of many variables.This strategy dramatically increases the efficiency of sequential multiscale modeling,making it very competitive in many situations.
基金supported by National Natural Science Foundation of China(Grant Nos.12071046,11571002,11772067).
文摘In this paper,a direct arbitrary Lagrangian-Eulerian(ALE)discontinuous Galerkin(DG)scheme is proposed for simulating compressible multi-material flows on the adaptive quadrilateral meshes.Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture.The coupled system is discretized in the reference element and we use a kind of Taylor expansion basis functions to construct the interpolation polynomials of the variables.We show the property that the material derivatives of the basis functions in the DG discretization are equal to zero,with which the scheme is simplified.In addition,the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from[H.Z.Tang and T.Tang,Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws,SIAMJ.NUMER.ANAL].This adaptivemesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation near the material interfaces in the simulations.With the help of this adaptivemesh method,the resolution of the solution near the target regions can be greatly improved and the computational efficiency of the simulation is increased.Our scheme can be applied in the simulations for the gas andwatermedia efficiently,and it ismore concise compared to some other methods such as the indirect ALE methods.Several examples including the gas-water flow problemare presented to demonstrate the efficiency of our scheme,and the results show that our scheme can capture the wave structures sharply with high robustness.
