This paper is concerned with Godunov＇s scheme for the initial-boundary value problem of scalar conservation laws. A kind of Godunov＇s scheme, which satisfies the boundary entropy condition, was given. By use of the ...This paper is concerned with Godunov＇s scheme for the initial-boundary value problem of scalar conservation laws. A kind of Godunov＇s scheme, which satisfies the boundary entropy condition, was given. By use of the scheme, numerical simulation for the weak entropy solution to the initial-boundary value problem of scalar conservation laws is conducted.展开更多
This paper investigates the effect of inflow, outflow and shock waves in a single lane highway traffic flow problem. A constant source term has been introduced to demonstrate the inflow and outflow. The classical Ligh...This paper investigates the effect of inflow, outflow and shock waves in a single lane highway traffic flow problem. A constant source term has been introduced to demonstrate the inflow and outflow. The classical Lighthill Whitham and Richards (LWR) model combined with the Greenshields model is used to obtain analytical and numerical solutions. The model is treated as an IBVP and numerical solutions are presented using Lax Friedrichs scheme. Godunov method is also used to present shock wave analysis. The numerical procedures adopted in this investigation yield results which are very much consistent with real life scenario in terms of traffic density and velocity.展开更多
This paper presents a cell-centered Godunov method based on staggered data distribu-tion in Eulerian framework.The motivation is to reduce the intrinsic entropy dissipation of classical Godunov methods in the calculat...This paper presents a cell-centered Godunov method based on staggered data distribu-tion in Eulerian framework.The motivation is to reduce the intrinsic entropy dissipation of classical Godunov methods in the calculation of an isentropic or rarefaction flow.At the same time,the property of accurate shock capturing is also retained.By analyzing the factors that cause nonphysical entropy in the conventional Godunov methods,we introduce two velocities rather than a single velocity in a cell to reduce kinetic energy dissipation.A series of redistribution strategies are adopted to update subcell quantities in order to improve accuracy.Numerical examples validate that the present method can dramatically reduce nonphysical entropy increase.Mathematics subject classification:35Q35,76N15,76M12.展开更多
The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attemp...The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attempt to cure this kind of non-physical phenomenon.In this paper,a matrix-based stability analysis for 2-D Euler equations is performed to explore the cause of instability of numerical schemes.By combining the Roe with HLL flux in different directions and different flux components,we give an interesting explanation to the linear numerical instability.Based on such analysis,some hybrid schemes are compared to illustrate different mechanisms in controlling shock instability.Numerical experiments are presented to verify our analysis results.The conclusion is that the scheme of restricting directly instability source is more stable than other hybrid schemes.展开更多
In this note,we propose a new method to cure numerical shock instability by hybriding different numerical fluxes in the two-dimensional Euler equations.The idea of this method is to combine a”full-wave”Riemann solve...In this note,we propose a new method to cure numerical shock instability by hybriding different numerical fluxes in the two-dimensional Euler equations.The idea of this method is to combine a”full-wave”Riemann solver and a”less-wave”Riemann solver,which uses a special modified weight based on the difference in velocity vectors.It is also found that such blending does not need to be implemented in all equations of the Euler system.We point out that the proposed method is easily extended to other”full-wave”fluxes that suffer from shock instability.Some benchmark problems are presented to validate the proposed method.展开更多
This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model.The HLL Riemann solver is applied to solve the Riemann problem for the model equations.This sol...This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model.The HLL Riemann solver is applied to solve the Riemann problem for the model equations.This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture.Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results.To demonstrate the robustness,effectiveness and capability of these methods,the model results are compared with reference solutions.In addition to that,these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature.The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.展开更多
Conservative numerical methods are often used for simulations of fluid flows involving shocks and other jumps with the understanding that conservation guarantees reasonable treatment near discontinuities.This is true ...Conservative numerical methods are often used for simulations of fluid flows involving shocks and other jumps with the understanding that conservation guarantees reasonable treatment near discontinuities.This is true in that convergent conservative approximations converge to weak solutions and thus have the correct shock locations.However,correct shock location results from any discretization whose violation of conservation approaches zero as the mesh is refined.Here we investigate the case of the Euler equations for a single gas using the Jones-Wilkins-Lee(JWL)equation of state.We show that a quasi-conservative method can lead to physically realistic solutions which are devoid of spurious pressure oscillations.Furthermore,we demonstrate that under certain conditions,a quasi-conservative method can exhibit higher rates of convergence near shocks than a strictly conservative counterpart of the same formal order.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No. 10671120）
文摘This paper is concerned with Godunov＇s scheme for the initial-boundary value problem of scalar conservation laws. A kind of Godunov＇s scheme, which satisfies the boundary entropy condition, was given. By use of the scheme, numerical simulation for the weak entropy solution to the initial-boundary value problem of scalar conservation laws is conducted.
文摘This paper investigates the effect of inflow, outflow and shock waves in a single lane highway traffic flow problem. A constant source term has been introduced to demonstrate the inflow and outflow. The classical Lighthill Whitham and Richards (LWR) model combined with the Greenshields model is used to obtain analytical and numerical solutions. The model is treated as an IBVP and numerical solutions are presented using Lax Friedrichs scheme. Godunov method is also used to present shock wave analysis. The numerical procedures adopted in this investigation yield results which are very much consistent with real life scenario in terms of traffic density and velocity.
基金supported by the National Natural Science Foundation of China(Grant Nos.11971071,12302377)by the Foundation of LCP(Grant No.6142A05220201)by the China Postdoctoral Science Foundation(Grant No.2022M722185).
文摘This paper presents a cell-centered Godunov method based on staggered data distribu-tion in Eulerian framework.The motivation is to reduce the intrinsic entropy dissipation of classical Godunov methods in the calculation of an isentropic or rarefaction flow.At the same time,the property of accurate shock capturing is also retained.By analyzing the factors that cause nonphysical entropy in the conventional Godunov methods,we introduce two velocities rather than a single velocity in a cell to reduce kinetic energy dissipation.A series of redistribution strategies are adopted to update subcell quantities in order to improve accuracy.Numerical examples validate that the present method can dramatically reduce nonphysical entropy increase.Mathematics subject classification:35Q35,76N15,76M12.
基金supported by the National Natural Science Foundation of China(11071025)the Foundation of CAEP(2010A0202010)the Foundation of National Key Laboratory of Science and Technology Computation Physics and the Defense Industrial Technology Development Program(B1520110011).
文摘The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attempt to cure this kind of non-physical phenomenon.In this paper,a matrix-based stability analysis for 2-D Euler equations is performed to explore the cause of instability of numerical schemes.By combining the Roe with HLL flux in different directions and different flux components,we give an interesting explanation to the linear numerical instability.Based on such analysis,some hybrid schemes are compared to illustrate different mechanisms in controlling shock instability.Numerical experiments are presented to verify our analysis results.The conclusion is that the scheme of restricting directly instability source is more stable than other hybrid schemes.
基金supported in part by the National Natural Science Foundation of China under(Grant No.10871029)foundation of LCP.
文摘In this note,we propose a new method to cure numerical shock instability by hybriding different numerical fluxes in the two-dimensional Euler equations.The idea of this method is to combine a”full-wave”Riemann solver and a”less-wave”Riemann solver,which uses a special modified weight based on the difference in velocity vectors.It is also found that such blending does not need to be implemented in all equations of the Euler system.We point out that the proposed method is easily extended to other”full-wave”fluxes that suffer from shock instability.Some benchmark problems are presented to validate the proposed method.
文摘This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model.The HLL Riemann solver is applied to solve the Riemann problem for the model equations.This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture.Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results.To demonstrate the robustness,effectiveness and capability of these methods,the model results are compared with reference solutions.In addition to that,these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature.The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.
基金supported by Lawrence Livermore National Laboratory under the auspices of the U.S.Department of Energy through contract number DE-AC52-07NA27344.
文摘Conservative numerical methods are often used for simulations of fluid flows involving shocks and other jumps with the understanding that conservation guarantees reasonable treatment near discontinuities.This is true in that convergent conservative approximations converge to weak solutions and thus have the correct shock locations.However,correct shock location results from any discretization whose violation of conservation approaches zero as the mesh is refined.Here we investigate the case of the Euler equations for a single gas using the Jones-Wilkins-Lee(JWL)equation of state.We show that a quasi-conservative method can lead to physically realistic solutions which are devoid of spurious pressure oscillations.Furthermore,we demonstrate that under certain conditions,a quasi-conservative method can exhibit higher rates of convergence near shocks than a strictly conservative counterpart of the same formal order.
