A compactness frame of the Lax-Friedrichs scheme for the equations of gas dynamics is obtained by using some embedding theorems and an analysis of the difference scheme and the weak entropy.
Derive L-2-error bounds for Lax-Friedrichs schemes for discontinuous solutions oflinear hyperbolic convection equations.It is known that the Lax-Friedrichs scheme is a firstorder scheme.Analyzes convergent rate of the...Derive L-2-error bounds for Lax-Friedrichs schemes for discontinuous solutions oflinear hyperbolic convection equations.It is known that the Lax-Friedrichs scheme is a firstorder scheme.Analyzes convergent rate of the scheme through its modified equations andshows that the first order Lax-Friedrichs scheme to approach BV solutions of the convectionequation has L￣2-error bounds of O(△x￣(1/4)),where △x is the discrete mesh length.Nemericalexperiments are presented and numerical results justify the theoretical analysis.展开更多
We give a brief discussion of some of the contributions of Peter Lax to Com- putational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 198...We give a brief discussion of some of the contributions of Peter Lax to Com- putational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We de- velop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax- Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh＇s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.展开更多
A fluid dynamic traffic flow model based on a non-linear velocity-density function is considered. The model provides a quasi-linear first order hyperbolic partial differential equation which is appended with initial a...A fluid dynamic traffic flow model based on a non-linear velocity-density function is considered. The model provides a quasi-linear first order hyperbolic partial differential equation which is appended with initial and boundary data and turns out an initial boundary value problem (IBVP). A first order explicit finite difference scheme of the IBVP known as Lax-Friedrich’s scheme for our model is presented and a well-posedness and stability condition of the scheme is established. The numerical scheme is implemented in order to perform the numerical features of error estimation and rate of convergence. Fundamental diagram, density, velocity and flux profiles are presented.展开更多
This paper is about the positivity analysis of a class of flux-vector splitting (FVS) methods for the compressible Euler equations, which include gas-kinetic Beam scheme[8], Steger-Warming FVS method[9], and Lax-Fried...This paper is about the positivity analysis of a class of flux-vector splitting (FVS) methods for the compressible Euler equations, which include gas-kinetic Beam scheme[8], Steger-Warming FVS method[9], and Lax-Friedrichs scheme. It shows that the density and the internal energy could keep non-negative values under the CFL condition for all above three schemes once the initial gas stays in a physically realizable state. The proof of positivity is closely related to the pseudo-particle representation of FVS schemes.展开更多
The artificial viscosity in the traditional smoothed particle hydrodynamics （SPH） methodology concerns some empirical coefficients, which limits the capability of the SPH methodology. To overcome this disadvantage a...The artificial viscosity in the traditional smoothed particle hydrodynamics （SPH） methodology concerns some empirical coefficients, which limits the capability of the SPH methodology. To overcome this disadvantage and further improve the accuracy of shock capturing, this paper introduces two other ways for numerical viscosity, which are the Lax-Friedrichs flux and the two- shock Riemann solver with MUSCL reconstruction to provide stability. Six SPH methods with different kinds of numerical viscosity are tested against the analytical solution for a 1-D dam break with a wet bed. The comparison shows that the Lax-Friedrichs flux with MUSCL reconstruction can capture the shock wave more accurate than other five methods. The Lax-Friedrichs flux and the artificial viscosity with MUSCL reconstruction are finally both applied to a 2-D dam-break test case in a L-shaped channel and the numerical results are compared with experimental data. It is concluded that this corrected SPH method can be used to solve shallow-water equations well.展开更多
In this paper,using the fractional step Lax-Friedrichs difference scheme,we establish the stability of the entropy solution on flow function and relaxation function for a class of conservation law systems with a sourc...In this paper,using the fractional step Lax-Friedrichs difference scheme,we establish the stability of the entropy solution on flow function and relaxation function for a class of conservation law systems with a source term and a relaxation term.展开更多
文摘A compactness frame of the Lax-Friedrichs scheme for the equations of gas dynamics is obtained by using some embedding theorems and an analysis of the difference scheme and the weak entropy.
文摘Derive L-2-error bounds for Lax-Friedrichs schemes for discontinuous solutions oflinear hyperbolic convection equations.It is known that the Lax-Friedrichs scheme is a firstorder scheme.Analyzes convergent rate of the scheme through its modified equations andshows that the first order Lax-Friedrichs scheme to approach BV solutions of the convectionequation has L￣2-error bounds of O(△x￣(1/4)),where △x is the discrete mesh length.Nemericalexperiments are presented and numerical results justify the theoretical analysis.
基金performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396supported in part by the Czech Science Foundation GrantP205/10/0814the Czech Ministry of Education grants MSM 6840770022 and LC528
文摘We give a brief discussion of some of the contributions of Peter Lax to Com- putational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We de- velop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax- Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh＇s infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.
文摘A fluid dynamic traffic flow model based on a non-linear velocity-density function is considered. The model provides a quasi-linear first order hyperbolic partial differential equation which is appended with initial and boundary data and turns out an initial boundary value problem (IBVP). A first order explicit finite difference scheme of the IBVP known as Lax-Friedrich’s scheme for our model is presented and a well-posedness and stability condition of the scheme is established. The numerical scheme is implemented in order to perform the numerical features of error estimation and rate of convergence. Fundamental diagram, density, velocity and flux profiles are presented.
文摘This paper is about the positivity analysis of a class of flux-vector splitting (FVS) methods for the compressible Euler equations, which include gas-kinetic Beam scheme[8], Steger-Warming FVS method[9], and Lax-Friedrichs scheme. It shows that the density and the internal energy could keep non-negative values under the CFL condition for all above three schemes once the initial gas stays in a physically realizable state. The proof of positivity is closely related to the pseudo-particle representation of FVS schemes.
基金Project supported by the National Natural Science Foun-dation of China(Grant No.51175001)the Natural Science Foundation of Anhui Province(Grant No.1508085QE100)the Higher Education of Anhui Provincial Scientific Research Project Funds(Grant No.TSKJ2015B03)
文摘The artificial viscosity in the traditional smoothed particle hydrodynamics （SPH） methodology concerns some empirical coefficients, which limits the capability of the SPH methodology. To overcome this disadvantage and further improve the accuracy of shock capturing, this paper introduces two other ways for numerical viscosity, which are the Lax-Friedrichs flux and the two- shock Riemann solver with MUSCL reconstruction to provide stability. Six SPH methods with different kinds of numerical viscosity are tested against the analytical solution for a 1-D dam break with a wet bed. The comparison shows that the Lax-Friedrichs flux with MUSCL reconstruction can capture the shock wave more accurate than other five methods. The Lax-Friedrichs flux and the artificial viscosity with MUSCL reconstruction are finally both applied to a 2-D dam-break test case in a L-shaped channel and the numerical results are compared with experimental data. It is concluded that this corrected SPH method can be used to solve shallow-water equations well.
文摘In this paper,using the fractional step Lax-Friedrichs difference scheme,we establish the stability of the entropy solution on flow function and relaxation function for a class of conservation law systems with a source term and a relaxation term.
