Lattice Boltzmann Method For Spatial Fractional Convection Diffusion Equations And Its Application

Fractional differential equations are widely focused and applied in the fields of complex physics,mechanics and environmental engineering,etc.However,there may be difficult to obtain the explicit form of exact solutions for the above equations in general because of the nonlocal characteristics of fractional operators.Therefore,it is particularly significant to study the numerical solutions of fractional differential equations.In this paper,based on the lattice Boltzmann method,a class of spatial fractional differential equations is numerically solved.Meanwhile,the fractional coupling model is established and applied in the simulation of heavy metal migration and transformation in deep-sea mining.A new lattice Boltzmann model is established to numerically solve the Caputo spatial fractional convection equation.Through the composite integration rule and linear interpolation method,the fractional derivative is discretized.The macroscopic equation which is required to be solved is recovered correctly and equilibrium distribution functions in each direction are derived by combining Taylor expansion and Chapman-Enskog multi-scale expansion.Numerical results verify the effectiveness of the constructed model via one-dimensional and two-dimensional problems.For spatial fractional convection-diffusion equation,a lattice Boltzmann model with correction term is established and solved numerically.By discretizing the integral terms of fractional order operator based on the integral mean value theorems,the fractional convection-diffusion equation is transformed into the standard one.Afterward,the convergence analysis is presented.In addition,the evolution equation with correction term is selected and a new lattice Boltzmann model is constructed.With Taylor expansion,Chapman-Enskog multi-scales expansion,the macroscopic equation which is required to be solved is recovered correctly.Furthermore,the equilibrium distribution functions of the established model are derived in all directions.Finally,for one-dimensional and two-dimensional problems,numerical experiments are carried out to verify the accuracy and efficiency of the present model by comparing examples with exact solutions.Based on the study of lattice Boltzmann method for fractional differential equations,the fractional coupling model is established and applied in the simulation of heavy metal migration and transformation in deep-sea mining.According to integer coupling model,the fractional coupling model considering the effects of convection-diffusion,adsorption-desorption and sedimentationresuspension process is established to describe the dissolution and transportation of heavy metal pollutants.Taking cadmium(Cd)as an example,lattice Boltzmann method is adopted to numerically solve the model.The concentration variation in the dissolved phase,suspended phase and sediment phase are analyzed in detail,and the comparison with the existing algorithms shows that LBM has higher computational efficiency.At the same time,the influence for the change of heavy metal pollutants,concentration as different α are discussed.The numerical results show the effectiveness of the present model.Finally,it’s worth noting that the various concentration of heavy metal pollutants with time and distance are discussed and analyzed combined with the water quality standard.