Numerical Methods For Fractional Partial Differential Equations And Generalized Phase Field Models

In this paper,we are focused on numerical solutions for time-fractional diffusion equations(TFDEs),time-fractional gradient flow,and a kind of generalized phase field model.The main contribution of this work is threefold:1)We propose an efficient numerical method for TFDEs.The main difficulty in solving this equation comes from the presence of the time fractional differential operator.Its non-locality makes both the storage and computation expensive,while making the stability analysis complicate.For the time discretization,we construct a fast(3-?)-order scheme for the Caputo fractional derivative of order a based on the L2 scheme and the sum-of-exponentials approximation to the convolution kernel involved in the fractional derivative.This work can be regarded as a continuation of previous works reported by[Lv&Xu,SISC 2016],in which a(3-?)-order time stepping scheme,known as L2 scheme now,was constructed and analyzed.We extend this scheme to take into account the fast evaluation method for the purpose to reduce the memory and overall computational cost.For the spatial discretization,we propose a spectral collocation method.Both bounded and unbounded domains are considered.The stability as well as the accuracy of the time-space full discrete problem are rigorously proven.Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.At last,we extend the discussion to graded mesh to make the scheme more suitable for problems having weakly singular solutions at the initial time.2)We investigate numerical solutions of some time-fractional phase field models.Based on an improved scalar auxiliary variable approach for gradient flows,we develop three different numerical schemes for the time-fractional Allen-Cahn equation,the time-fractional Cahn-Hillard equation,and the time-fractional molecular beam epitaxy model respectively.The stability of these schemes is proved by establishing the assocaited energy dissipation law.Some numerical tests are performed at last to demonstrate the efficiency of the proposed schemes.3)A modified higher-order(in space)anisotropic generalized Cahn-Hilliard model is studied.This kind of equations frequently arises in the study in biology,image processing,etc.We first deduce a priori estimates to prove well-posedness results and the dissipativity of the semigroup,as well as the existence of global attractor.Then we study a fully discrete scheme which is a combination of the finite element or spectral method in space and a second-order stable scheme in time.We obtain the stability result,as well as the existence and uniqueness of the numerical solution,both for the space semi-discrete and fully discrete cases.Finally,several numerical examples are provided to confirm the theoretical results.In particular,the effects of the higher-order terms on the anisotropy are numerically observed.