Numerical Solution And Algorithm Optimization Of Anomalous Dynamic Model

Anomalous diffusions are ubiquitous in the natural world,fractional calculus op-erators play an important role in describing the anomalous diffusion phenomena.But their non-locality,singularity and space-time coupling make the theoretical analy-ses and numerical algorithms for the fractional partial differential equation(system)immature,so how to solve the corresponding fractional partial differential equation-s(FPDEs)effectively is still a meaningful work.In this paper,we mainly discuss some specific FPDEs that describe the anomalous dynamic,including the regulari-ty analysis-the construction of numerical scheme and the corresponding algorithm optimization.The main contents of this paper are as followsIn the first chapter-we state the research background and status for fractional diffusion equation(system);at the same time,we introduce the main works and innovations of this paperIn the second chapter,we mainly discuss the regularity estimate and numerical solution for the backward fractional Feynman-Kac equation.We first provide the regularity estimate for this equation.And then,we use the backward Euler(BE)con-volution quadrature to discretize the Riemann-Liouville fractional substantial deriva-tive and finite element for space Laplace operator,and the effect of the time-space coupling of the fractional substantial derivative operator on the approximation ac-curacy can be weakened by adjusting the projection of the finite element scheme appropriately,so we provide the BE fully-discrete scheme for the backward fractional Feynman-Kac equation with nonsmooth data.At last,we use the convolution quadra-ture generated by high-order backward difference formula(BDF)to approximate the Riemann-Liouville fractional substantial derivative under the premise of the optimal convergence in space,then the convergence in time can achieve up to order 6 after making some corrections at each time stepIn the third chapter,we first provide the fast algorithms for convolution quadra-ture of the Riemann-Liouville fractional derivative,including the backward Euler(BE)and second-order backward difference(SBD)convolution quadrature,and then we use the fast algorithm to solve the homogeneous fractional Fokker-Planck system.The main advantage of our fast algorithm is that there is no assumption on the regularity of the solution in time.As we all know,the non-locality of the fractional derivative operator leads to a large amount of computation time and memory,especially for the large-scale system.In this chapter,we use the sum of geometric sequences to ap-proximate the weights generated by BE and SBD convolution quadrature effectively,so the computation can be performed iteratively and the computational complexity is reduced greatly.Furthermore,we know clearly how fast algorithm influence on accuracy from the error analysis.At last,we verify the effectiveness of our algorithm by the comparative numerical experiments.In the fourth chapter,we solve the two-dimensional space fractional diffusion equation numerically by the central local discontinuous Galerkin(CLDG)method.To the best of our knowledge,this method is mainly used to solve the integer-order partial differential equations,and we extend this method to solve the FPDEs.Compared with the traditional discontinuous Galerkin(DG),CLDG method use the duplicative information on overlapping cells and avoid using the numerical flux to define the interface values of the solution,and this scheme also covers the two main advantages of DG method:the flexibility of grid subdivision and excellent parallel efficiency.To guarantee the stability and convergence of our scheme,we modify the local DG scheme according to the theoretical prediction and the feature of the CLDG.Moreover,we provide the stability and error analysis for the discretization and the numerical experiments verify the effectiveness of our scheme.The provided numerical algorithm and theoretical analysis can be easily applied to the one-dimensional space fractional diffusion equation.In the fifth chapter,we make the split of the integral fractional Laplacian as(-?)su=(-?)(-?)s-1u,where s ?(0,1/2)?(1/2,1).Based on this splitting,we re-spectively discretize the one-and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding trunca-tion errors by the theory of the approximation.Moreover,the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an O(h1+?-2s)convergence rate can be obtained when the so-lution u ?C1,?(?n?)where n is the dimension of the space,? ?(max(0,2s-1),1],? is a fixed positive constant,and h denotes mesh size.Finally,extensive numerical experiments verify the effectiveness of the theoretical results.In the last chapter,we conclude our paper with some expectations for the future research work.