| In this paper,the time-fractional diffusion equation by fractional linear multistep methods(FLMMs)are studied,And we do the corresponding error analysis.Firstly,the fractional linear multistep method of time fractional diffusion equation is mainly studied.The space direction is approximated by finite element method,and two fully discrete schemes are obtained.Theoretical analysis shows that the two methods are unconditionally stable,and the optimal error estimates in the sense of L2 norm in space are also obtained.For the smooth solution,the convergence order is O(τ2-β)at time direction away from t=0.For smooth initial values and source terms(non-smooth solutions),the convergence order in time direction is O(τ1+β)The results of theoretical analysis are verified by numerical experiments.Secondly,for the problem of non-smooth solution,the improved algorithm of fractional linear multistep scheme is adopted.By adding some appropriate correction items in the discrete time direction and using finite element method in the space direction,the scheme can overcome the influence of the low regularity of the solution in time and has higher precision and convergence order.The energy analysis method is used to prove the absolute stability of the numerical scheme.For the non-smooth solution,the convergence order in time direction is improved to a certain extent.The results of theoretical analysis are verified by numerical experiments. |
