Second-order Convergence Rate In Space Research Articles

Macroscopic finite-difference scheme and modified equationsof the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system-based incompressible Navier-Stokes equations.

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equationswhich are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equationscan be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity. Finally, we also conduct some simulations to test the MDF-LBM and find that the numerical results not only agree with some available analytical and numerical solutions but also have a second-order convergence rate in space.

A lattice Boltzmann model for the conjugate heat transfer

Conjugate heat transfer problems are of great importance in many applications, while the available lattice Boltzmann (LB) models have some limitations in the study of such problems with the heterogeneous media. In this paper, we develop a new LB model for the conjugate heat transfer problems where the evolution equation with a modified equilibrium distribution function and a source term is considered. Through the Chapman–Enksog analysis, the macroscopic temperature equation can be recovered correctly. In order to test the accuracy, efficiency and stability of this LB model, some simple but nontrivial benchmark problems, including the planar thermal Poiseuille flow with two immiscible fluids, the conjugate heat transfer between a stratified heterogeneous medium with variable thermophysical properties, the steady and unsteady-state planer thermal flows with the two immiscible fluids, the steady-state conjugate heat transfer problem with a vertical interface, and the natural convection in a square cavity filled with circular and square rods, are studied. It is found that the results obtained from the present LB model are in good agreement with the analytic solutions or some previous works. In addition, it is also demonstrated that the developed LB model has a second-order convergence rate in space, and is more stable than some previous LB models. Finally, it should be noted that in our work, there is no special treatment to the interface, and the continuity conditions at the interface can be satisfied automatically.

A lattice Boltzmann model for the nonlinear thermistor equations

In this paper, we propose a general and efficient lattice Boltzmann (LB) model for solving the nonlinear thermistor equations, where the nonlinear diffusion and Poisson equations are solved by two LB equations. Through Chapman–Enskog analysis, the nonlinear thermistor equations can be recovered correctly from the present LB model. We then test the model through some numerical simulations, and find that the numerical results are in good agreement with analytical solutions. Additionally, the numerical results also show that the present LB model has a second-order convergence rate in space.

Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers’ Equation**Supported by the National Natural Science Foundation of China under Grant No. 51576079

In this paper, we present a Cole-Hopf transformation based lattice Boltzmann (LB) model for solving one-dimensional Burgers’ equation, and compared to available LB models, the effect of nonlinear convection term can be eliminated. Through Chapman-Enskog analysis, it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB model. Some numerical tests are also performed to validate the present LB model, and the numerical results show that, similar to previous LB models, the present model also has a second-order convergence rate in space, but it is more accurate than the previous ones.

A comparative study of local and nonlocal Allen-Cahn equations with mass conservation

The local and nonlocal Allen-Cahn equations (ACEs) have received increasing attention in the study of the complicated interfacial problems. In this paper, we conduct a comparison between local and nonlocal ACEs with the property of mass conservation in the framework of lattice Boltzmann (LB) method. To this end, we first propose two simple multiple-relaxation-time LB models for local and nonlocal ACEs, and through the Chapman-Enskog expansion, the local and nonlocal ACEs can be recovered correctly from the developed LB models. Then we test these two LB models with several examples, and the numerical results show that the developed LB models are accurate and also have a second-order convergence rate in space. Finally, a comparison between the local and nonlocal ACEs is also performed in terms of mass conservation, stability and accuracy. The results show that both local and nonlocal ACEs can preserve mass conservation of system and each phase. And additionally, it is also found that the local ACE is more accurate than nonlocal ACE in capturing the interface profile, but the latter is more stable than the former.

Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.

Consistent forcing scheme in the cascaded lattice Boltzmann method.

In this paper, we give an alternative derivation for the cascaded lattice Boltzmann method (CLBM) within a general multiple-relaxation-time (MRT) framework by introducing a shift matrix. When the shift matrix is a unit matrix, the CLBM degrades into an MRT LBM. Based on this, a consistent forcing scheme is developed for the CLBM. The consistency of the nonslip rule, the second-order convergence rate in space, and the property of isotropy for the consistent forcing scheme is demonstrated through numerical simulations of several canonical problems. Several existing forcing schemes previously used in the CLBM are also examined. The study clarifies the relation between MRT LBM and CLBM under a general framework.

Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations

In this paper, a finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed. In this model, the equilibrium distribution function is delicately designed in order to recover the convection-diffusion equation exactly. Different from the standard LB model, the temporal and spatial steps in this model are decoupled such that it is convenient to study convection-diffusion problem with the non-uniform grid. In addition, it also preserves the advantage of standard LB model that the complex-valued convection-diffusion equation can be solved directly. The von Neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model. To test the performance of the model, a series of numerical simulations of some classical problems, including the diffusion equation, the nonlinear heat conduction equation, the Sine-Gordon equation, the Gaussian hill problem, the Burgers–Fisher equation, and the nonlinear Schrödinger equation, have also been carried out. The results show that the present model has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.

Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations.

In this paper, a comparative study of the lattice Boltzmann (LB) models for the Allen-Cahn (A-C) and Cahn-Hilliard (C-H) equations is conducted. To this end, a new LB model for the A-C equation is first proposed, where the equilibrium distribution function and the source term distribution function are delicately designed to recover the A-C equation correctly. The gradient term in this model can be computed by the nonequilibrium part of the distribution function such that the collision process can be implemented locally. Then a detailed numerical study on several classical problems is performed to give a comparison between the present model for the A-C equation and the recently developed LB model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)PLEEE81539-375510.1103/PhysRevE.89.053320] for the C-H equation in terms of tracking the interface of two-phase flow. The results show that the present LB model for the A-C equation is more accurate and more stable, and also has a second-order convergence rate in space, while the convergence rate of the previous LB model for the C-H equation is only about 1.5.

Lattice Boltzmann model for a steady radiative transfer equation.

A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for incompressible thermal flows

In this paper, a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann (LB) method for simulating incompressible thermal flows is presented. In the method, the incompressible Navier–Stokes equations and temperature equation are solved separately by two different MRT-LB equations, which are developed based on non-orthogonal basis vectors obtained from the combinations of the lattice velocity components. The macroscopic governing equations of incompressible thermal flows can be recovered from the method through the Chapman–Enskog analysis in the incompressible limit. Numerical simulations of several typical two-dimensional problems are carried out to validate the proposed method. It is found that the present results are in good agreement with the analytical solutions and/or other numerical results reported in the literature. Furthermore, the non-orthogonal MRT-LB model shows better numerical stability in comparison with the BGK-LB model, and the grid convergence tests indicate that the present MRT-LB method has a second-order convergence rate in space.

Discrete effect on the halfway bounce-back boundary condition of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations.

In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.

A Multiple-Relaxation-Time Lattice Boltzmann Model for General Nonlinear Anisotropic Convection–Diffusion Equations

In this paper, based on the previous work (Shi and Guo in Phys Rev E 79:016701, 2009), we develop a multiple-relaxation-time (MRT) lattice Boltzmann model for general nonlinear anisotropic convection---diffusion equation (NACDE), and show that the NACDE can be recovered correctly from the present model through the Chapman---Enskog analysis. We then test the MRT model through some classic CDEs, and find that the numerical results are in good agreement with analytical solutions or some available results. Besides, the numerical results also show that similar to the single-relaxation-time lattice Boltzmann model or so-called BGK model, the present MRT model also has a second-order convergence rate in space. Finally, we also perform a comparative study on the accuracy and stability of the MRT model and BGK model by using two examples. In terms of the accuracy, both the analysis and numerical results show that a numerical slip on the boundary would be caused in the BGK model, and cannot be eliminated unless the relaxation parameter is fixed to be a special value, while the numerical slip in the MRT model can be overcome once the relaxation parameters satisfy some constrains. The results in terms of stability also demonstrate that the MRT model could be more stable than the BGK model through tuning the free relaxation parameters.

Second-order lattice Boltzmann methods for PDEs of Asian option pricing with regime switching

This paper establishes lattice Boltzmann models with five amending functions for solving system of partial differential equations (PDEs) arising in Asian options pricing with regime switching. With the Chapman–Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and then the lattice Boltzmann method (LBM) is proposed. In the LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants in the traditional LBMs. The LBM has second-order convergence rate in space and first-order convergence rate in time. The stability, convergence rates and computational cost of LBMs are studied and verified by numerical examples.

A novel lattice Boltzmann model for the coupled viscous Burgers’ equations

In this work, a new lattice Boltzmann model for coupled Burgers’ equations is proposed through selecting proper distribution functions. Unlike the previous models, the present model can exactly recover the coupled equations without any assumptions. A detailed numerical study on several coupled Burgers’ equations is performed to validate the present model, and the results show that the present model not only has a second-order convergence rate in space, but also is more accurate than the previous model.

Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations.

In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = 10(12)).

Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method.

In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.

Lattice Boltzmann model for the convection-diffusion equation

We propose a lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) and show that the CDE can be recovered correctly from the model by the Chapman-Enskog analysis. The most striking feature of the present LB model is that it enables the collision process to be implemented locally, making it possible to retain the advantage of the lattice Boltzmann method in the study of the heat and mass transfer in complex geometries. A local scheme for computing the heat and mass fluxes is then proposed to replace conventional nonlocal finite-difference schemes. We further validate the present model and the local scheme for computing the flux against analytical solutions to several classical problems, and we show that both the model for the CDE and the computational scheme for the flux have a second-order convergence rate in space. It is also demonstrated the present model is more accurate than existing LB models for the CDE.