The thermal lattice Boltzmann equation(TLBE)has superior numerical stability as an explicit algorithm.However,its applications on nonuniform meshes are complicated.This paper clarifies the intrinsic mechanism for stab...The thermal lattice Boltzmann equation(TLBE)has superior numerical stability as an explicit algorithm.However,its applications on nonuniform meshes are complicated.This paper clarifies the intrinsic mechanism for stabilizing computations in TLBE and proposes two solvers that combine the numerical stability of TLBE and flexible finite difference/volume schemes for nonuniform meshes.Through a brief review of the lattice Boltzmann method,it is concluded that the entropy increase of the collision operator is essential for numerical stability.This paper first proposes a macroscopic entropy-increasing(MEI)model for convection-diffusion problems by combining the MEI process and TLBE.The von Neumann stability analysis proves that the MEI model has no upper limit for mesh Fourier number.However,the accuracy of the MEI model is found to be sensitive to higher-order deviation terms.Therefore,a hybrid model that combines the MEI and the equilibrium-moment-based models is proposed to solve the problem.The von Neumann stability analysis demonstrates that the hybrid model can completely recover the numerical stability of TLBE.Numerical investigations validate the good stability and accuracy of the hybrid model.Most importantly,it can be easily applied to nonuniform meshes,whereas implementing TLBE on nonuniform meshes is relatively complicated.展开更多
The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general dif...The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general difference schemes with unequal meshsteps are established by the fixed point technique. The absolute and relative convergence of the discrete vector solution are justified by a series of a priori estimates. The analysis of mentioned problems are based on the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem of the nonlinear parabolic system.展开更多
In this paper, a difference scheme with nonuniform meshes is established for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in spaces...In this paper, a difference scheme with nonuniform meshes is established for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in spacestep and timestep.展开更多
In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order converge...In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.展开更多
An investigation of postshock oscillations on non-uniform grids is performed in this paper. These oscillations are generated as shock passes through the grid interfaces. The LLF scheme is checked for 1D and 2D problem...An investigation of postshock oscillations on non-uniform grids is performed in this paper. These oscillations are generated as shock passes through the grid interfaces. The LLF scheme is checked for 1D and 2D problems on the discontinuous grids. Oscillations are observed only for nonlinear systems and the solutions of the scalar conservation laws and linear systems behave logically. The integral curves suggest underlying properties of these oscillations. The results of the paper reveal a flaw that adaptive methods for conservation laws have to refine grids at each time step.展开更多
In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmo...In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.展开更多
In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both spac...In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.展开更多
基金support of the Alexander von Humboldt Foundation,Germany.
文摘The thermal lattice Boltzmann equation(TLBE)has superior numerical stability as an explicit algorithm.However,its applications on nonuniform meshes are complicated.This paper clarifies the intrinsic mechanism for stabilizing computations in TLBE and proposes two solvers that combine the numerical stability of TLBE and flexible finite difference/volume schemes for nonuniform meshes.Through a brief review of the lattice Boltzmann method,it is concluded that the entropy increase of the collision operator is essential for numerical stability.This paper first proposes a macroscopic entropy-increasing(MEI)model for convection-diffusion problems by combining the MEI process and TLBE.The von Neumann stability analysis proves that the MEI model has no upper limit for mesh Fourier number.However,the accuracy of the MEI model is found to be sensitive to higher-order deviation terms.Therefore,a hybrid model that combines the MEI and the equilibrium-moment-based models is proposed to solve the problem.The von Neumann stability analysis demonstrates that the hybrid model can completely recover the numerical stability of TLBE.Numerical investigations validate the good stability and accuracy of the hybrid model.Most importantly,it can be easily applied to nonuniform meshes,whereas implementing TLBE on nonuniform meshes is relatively complicated.
文摘The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general difference schemes with unequal meshsteps are established by the fixed point technique. The absolute and relative convergence of the discrete vector solution are justified by a series of a priori estimates. The analysis of mentioned problems are based on the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem of the nonlinear parabolic system.
文摘In this paper, a difference scheme with nonuniform meshes is established for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in spacestep and timestep.
文摘In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.
文摘An investigation of postshock oscillations on non-uniform grids is performed in this paper. These oscillations are generated as shock passes through the grid interfaces. The LLF scheme is checked for 1D and 2D problems on the discontinuous grids. Oscillations are observed only for nonlinear systems and the solutions of the scalar conservation laws and linear systems behave logically. The integral curves suggest underlying properties of these oscillations. The results of the paper reveal a flaw that adaptive methods for conservation laws have to refine grids at each time step.
文摘In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.
基金Supported by the National Natural Science Foundation of China(No.10671060,No.10871061)the Youth Foundation of Hunan Education Bureau(No.06B037)the Construct Program of the Key Discipline in Hunan Province
文摘In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.
