求解三维空间分数阶对流扩散方程的Douglas-Gunn格式
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摘要
由于分数阶导数的非局部性特征,在模拟反常扩散现象时使用分数阶偏微分方程具有更好的效果,但是分数阶导数的非局部性也给数值分析和计算带来了很大困难,尤其在多维空间情形下.通过对经典Douglas-Gunn格式的推广,提出一种求解三维空间分数阶对流扩散方程(space fractional advection diffusion equation,SFADE)的交替方向隐(alternating direction implicit,ADI)差分格式,并用矩阵法证明了其稳定性和收敛性.用数值算例进一步验证了该格式在空间和时间方向均具有较高的二阶收敛精度,可以高效地求解三维SFADE.
Due to the non-locality of fractional derivatives,fractional partial differential equations were better to describe anomalous diffusion phenomena than other methods. However,while enjoying the convenience from mathematical modeling,it also caused lots of trouble especially in solving multidimensional cases. An efficient numerical algorithm was proposed for solving the three-dimensional space fractional advection diffusion equation(SFADE) by generalizing the Douglas-Gunn scheme. Stability and convergence of the method were proved by the matrix method. The derived alternating direction implicit(ADI)finite difference scheme had the second order accuracy in both time and space directions,respectively.The efficiency and convergence orders were finally demonstrated by some numerical examples.
Due to the non-locality of fractional derivatives,fractional partial differential equations were better to describe anomalous diffusion phenomena than other methods. However,while enjoying the convenience from mathematical modeling,it also caused lots of trouble especially in solving multidimensional cases. An efficient numerical algorithm was proposed for solving the three-dimensional space fractional advection diffusion equation(SFADE) by generalizing the Douglas-Gunn scheme. Stability and convergence of the method were proved by the matrix method. The derived alternating direction implicit(ADI)finite difference scheme had the second order accuracy in both time and space directions,respectively.The efficiency and convergence orders were finally demonstrated by some numerical examples.
引文
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[5] HEJAZI H,MORONEY T,LIU F. Stability and convergence of a finite volume method for the space fractional advection-dispersion equation[J]. Journal of computational and applied mathematics,2014,255:684-697.
[6]虎晓燕,韩惠丽.重心插值配点法求解分数阶Fredholm积分方程[J].郑州大学学报(理学版),2017,49(1):17-23.
[7] ZHENG M,LIU F,ANH V,et al. A high-order spectral method for the multi-term time-fractional diffusion equations[J]. Applied mathematical modelling,2016,40(7/8):4970-4985.
[8] DENG W H,CHEN M H. Efficient numerical algorithms for three-dimensional fractional partial differential equations[J].Journal of computational mathematics,2014,32(4):371-391.
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[10] WANG H,DU N. Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations[J]. Journal of computational physics,2014,258:305-318.
[11] CHEN M H,DENG W H. A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation[J]. Applied mathematical modelling,2014,38(13):3244-3259.
[12] LAUB A J. Matrix analysis for scientists and engineers[M]. Philadelphia:SIAM,2005.
[13] QUARTERONI A,SACCO R,SALERI F. Numerical mathematics[M]. Berlin:Springer-Verlag,2007.
[2] ZHANG H,LIU F,ZHUANG P,et al. Numerical analysis of a new space-time variable fractional order advection-dispersion equation[J]. Applied mathematics and computation,2014,242:541-550.
[3] ERVIN V J,HEUER N,ROOP J P. Numerical approximation of a time dependent,nonlinear,space-fractional diffusion equation[J]. SIAM journal on numerical analysis,2007,45(2):572-591.
[4] ZHENG Y Y,LI C P,ZHAO Z G. A note on the finite element method for the space fractional advection diffusion equation[J]. Computers and mathematics with applications,2010,59:1718-1726.
[5] HEJAZI H,MORONEY T,LIU F. Stability and convergence of a finite volume method for the space fractional advection-dispersion equation[J]. Journal of computational and applied mathematics,2014,255:684-697.
[6]虎晓燕,韩惠丽.重心插值配点法求解分数阶Fredholm积分方程[J].郑州大学学报(理学版),2017,49(1):17-23.
[7] ZHENG M,LIU F,ANH V,et al. A high-order spectral method for the multi-term time-fractional diffusion equations[J]. Applied mathematical modelling,2016,40(7/8):4970-4985.
[8] DENG W H,CHEN M H. Efficient numerical algorithms for three-dimensional fractional partial differential equations[J].Journal of computational mathematics,2014,32(4):371-391.
[9] CHEN J,LIU F,LIU Q,et al. Numerical simulation for the three-dimension fractional sub-diffusion equation[J]. Applied mathematical modelling,2014,38(15):3695-3705.
[10] WANG H,DU N. Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations[J]. Journal of computational physics,2014,258:305-318.
[11] CHEN M H,DENG W H. A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation[J]. Applied mathematical modelling,2014,38(13):3244-3259.
[12] LAUB A J. Matrix analysis for scientists and engineers[M]. Philadelphia:SIAM,2005.
[13] QUARTERONI A,SACCO R,SALERI F. Numerical mathematics[M]. Berlin:Springer-Verlag,2007.