| The traditional calculus theories and models are the most commonly used math?ematical theories and models for us to carry out theoretical research,describe natural phenomena and guide industrial application.We have applied these mod-els and the development numerical simulations to understand the world better,promote industrial production,and facilitate our lives.In recent years,a series of studies have found that the classical mathematical models with the theoret-ical framework on a lot of problems and phenomena cannot provide accurate description of the phenomenon which can be seen everywhere in nature,such as anomalous super diffusion,fracture problems,memory and heredity,random bouncing nonlocal diffusion and so on.Therefore,it is necessary to break through the limitation of traditional models to develop new mathematical theories and related models.With the rapid development of non-local calculus and fraction-al calculus,the theoretical research and application of non-local problems have been paid much attention.What's more,these non-local models can simulate these non-local effects better.More importantly,the local models can be derived formally as special cases of the non-local models.Therefore,the non-local model can be seen as a generalization of the local model.The non-local terms includ-ed in the non-local model are generally composed of non-local vector operators,fractional derivatives,fractional integrals and so on.Although the development of non-local models has been relatively long,non-local vector operators have only been proposed in recent years.Prof.Du et.al.[1]defined a series of the local vector operators,including the nonlocal divergence operator,nonlocal gradient operator,nonlocal curl operator and their adjoint operators during the period of studying random bouncing nonlocal diffusion models in 2013.However,the de-velopment of fractional derivative and fractional integral have experienced a long period of trial and error.In fact,the beginning of fractional calculus and the emergence of classical calculus axe almost at the same time.But it wasn't found the true colors until the mathematician Euler discovered the gamma function.During this period,mathematicians Laplace,Riemann,Liouville,Letnikov,Rietz and others have made important contributions.Currently,the commonly used definitions of fractional derivatives are Caputo derivatives,Riemann-Liouville derivatives,Grunwald-Letnikov derivatives,Caputo-Fabrizio derivatives,and so on.At present,some models which we often used are nonlocal diffusion model,peridynamics model,time fractional diffusion model,space fractional convection diffusion model,nonlocal and fractional phase field model and so on.Nowa-days,non-local theories have been widely used in continuous mechanics,fracture mechanics,quantum mechanics,physics,materials science,economics,image pro-cessing and many other aspects.However,due to the influence of non-local terms,the resulting linear system often has non-local characteristics in the numerical simulation of non-local model-s.For example,if we consider the finite element method to discrete peridynamics model,the coefficient matrix of linear systems will be a dense matrix under the condition of ?>>h.It will require huge computation and storage to solve this matrix equation.For the linear system obtained by numerical discretization for solving fractional models,it also needs the huge demand of computational quan-tity and storage capacity.This requests us to find fast procedures to compute the linear system more quickly and more effective under the condition of fixed computer memory and calculation speed which will direct the industrial produc-tion more timely and accurately.So many challenges in computing speed make people to consider the fast procedures.In this paper,we will focus on this aspect and try to find a suitable numerical simulation method to solve the linear system generated by numerical format quickly and effectively.We focus on some classical non-local models such as steady state peridynamics models,time fractional mod-els and fractional phase field models.The numerical discrete methods are mainly the finite element method and finite difference method.For fast calculation,the method is mainly based on fast Fourier transform method to solve matrix vector multiplication.In detail,In chapter 1,We give brief introductions of the definition,background and development of non-local calculus,fractional calculus,time fractional models and fractional phase field model to facilitate the study of later specific models.In chapter 2,We mainly consider the numerical simulation and fast proce-dures for stable one-dimensional peridynamics model.The peridynamics model is a non-local model which is proposed by Prof.Silling in Sandia national labo-ratory in 2000 for the study of discontinuous long distance forces[4].At present,the peridynamics model has been successfully applied to the static and dynamic simulation,fracture,failure analysis of different materials and structures.The computational work and memory requirement are bottlenecks for numerical dis-crete methods for peridynamic models because of their non-local characteristics caused by the non-local influence domain constant ?.Therefore,we propose fast Galerkinand hp-Galerkin finite element methods to solve this model quickly.First,in view of the continuous peridynamics model,we consider a fast com-puting procedure to calculate the linear systems obtained by piecewise linear,piecewise quadratic,and piecewise cubic Galerkin methods.This fast solution technique is based on a fast Fourier transform and depends on the special struc-ture of coefficient matrices,and it helps to reduce the computational work from O(N3)required by traditional methods to 0(Nlog2N)and the memory require ment from 0(N2)to O(N)without using any lossy compression,where N is the number of unknowns.Secondly,The peridynamic model admits solutions having jump discontinuities by replacing differentiation with integration,so that the pro-posed method can be successfully applied.For problems with discontinuous solu-tions,we therefore introduce a piecewise-constant Galerkin method and give an h-and p-refinement algorithm.Then we develop fast hp-Galerkin methods based on hybrid piecewise-constant/piecewise-linear and piecewise-constant/piecewise-quadratic finite element approximations.The new method reduces the computa-tional work from O(N3)required by the traditional methods to O(N2)and the memory requirement from O(N2)to O(N).Finally,some numerical examples are given to verify the correctness and effectiveness of our fast algorithm.In chapter 3,we consider the numerical discrete formulations for time frac-tional ordinary and partial differential equations with Caputo fractional deriva-tives and the fast algorithms for the derived linear system.The fast algorithms for the space fractional models are often based on the Toeplitz property of coef-ficient matrix and the fast Fourier transform method to reduce the computation and storage capacity.Even though the coefficient matrix is sparse matrix of the numerical discrete formulation for time fractional model,the values at the new time layer will need all values of the old time layers which become the difficulty of fast algorithm.It will cost much calculation time any memory storage.More importantly,the coefficient.matrix has no Toeplitz property,so,the method of fast Fourier transform will be invalid.Therefore,the development of fast algo-rithm for time fractional equation is not as complete as that of space fractional equation.We consider different computing procedures to obtain the numerical solutions quickly.Firstly,a series of finite difference discretizations are consid-ered to solve the initial boundary value problems of time fractional equations.The corresponding fast algorithms axe considered for the derived linear systems.These fast algorithms are based on fast Fourier transform and transform of solu-tion order.For a fast solution technique to solve a two-sided fractional ordinary differential equation,it helps to reduce the computational work from O(N3)re-quired by traditional methods to O(Nlog2N)and the memory requirement from O(N2)to O(N)without using any lossy compression,where N is the number of unknowns.For time fractional convection equations with different fractional order ?,a fast solution technique depending on the special structure of coefficient matrices by rearranging the order of unknowns in space and time directions.It helps to reduce the computational work from O(MN2)required by traditional methods to O(NMlog2N)and the memory requirement from O(NM)to O(N)without using any lossy compression,where N = T-1 and-r is the size of time step,M = h-1 and h is the size of space step.For classical time fractional diffusion model,we consider the classic L1 discrete finite difference scheme and the finite element method with smooth and nonsmooth initial data.We solve the numerical solutions at all time layers by changing the approximate solution order.By this procedure,fast Fourier transform has been successfully applied to the matrix-vector multiplication.Importantly,a fast method is employed to solve the clas-sical time fractional diffusion equation with a lower cost at O(MNlog2N),where the direct method requires a1 overall computational complexity of O(MN2).In addition,we consider a high order compact finite difference numerical solution for the fractional Cable equation.We give strict error analysis,stability analysis and fast algorithm.Finally,some numerical examples are given to verify the correctness and effectiveness of our theoretical analysis.In chapter 4,we consider the unconditional energy stability scheme for the non-local phase field model and relative fast solution method.The phase field model was originally proposed to avoid the difficulty of tracking the liquid-solid interface in solidification tissue simulation.Nowadays,phase field model has been applied to mathematics,mechanics,materials science and other fields.What's more,it has been successfully applied to deal with a variety of situations of incompressible two-phase flow problems.Recently,non-local phase field model such as Cahn-Hilliard model with space nonlocal operator[84],space fractional Cahn-Hilliard model[85],the space-time fractional Allen-Cahn model[86],time fractional Cahn-Hilliard model[87,88]and so on has attracted more and more people's interest,and it has been applied to the design,material science,physics,image processing,and many aspects of economics.The main goal of this paper is to construct accurate and efficient linear algorithms for the general nonlocal Cahn-Hilliard equation with general nonlinear potential and prove the uncondi-tional energy stability for its semi-discrete schemes carefully and rigorously.We construct and analyze linear,first and second order(in time)numerical scalar auxiliary variable approaches to construct unconditionally energy stable schemes.In addition,considering the huge computational work and memory requirement in solving the linear system,we analyse the structure of the stiffness matrix and seek some effective fast solution method to reduce the computational work and memory requirement.By applying four transformation operators A1,A2,A3,A4,we transform the stiffness matrix into a block-Toeplitz-Toeplitz-block(BT-TB)matrix.Then,a fast solution technique which is based on a fast Fourier transform is presented to solve a new linear system with BTTB stiffness matrix.The overall colputational cost of the fast conjugate method is O(Nlog2N),since the number of iterations is O(logN)where N is the number of unknowns.What we need to focus is that if one use the Gaussian elimination method straight-forwardly to this linear system,then it requires O(N3)complexity.In addition,since N x N BTTB matrix is determined by only 2N-1 entries rather than N2 entries,the fast solver will make memory requirement from(O(N2)to O(N).Fi-nally,various 2D numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.In chapter 4,we consider the unconditional energy stability scheme for the time fractional phase field model.In this chapter,we mainly study two classi-cal phase field models including Caputo time fractional derivative,namely time fractional order Cahn-Hilliard model and time fractional Allen-Cahn model.The main goal is to construct accurate and efficient linear algorithms for the time frac-tional Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential and prove the unconditional energy stability for the time fractional Cahn-Hilliard and Allen-Cahn model and its semi-discrete schemes carefully and rigorously.Schemes with this property is extremely preferred for solving diffusive systems due to the fact that it is not only critical for the numerical scheme to capture the correct long time dynamics of the system,but also supplies sufficient flexi-bility for dealing with the stiffness issue.Moreover,the noncompliance of energy dissipation laws may lead to spurious numerical approximations if the mesh or time step size are not controlled carefully.However,due to the thin interface,it is a quite difficult issue to construct unconditionally energy stable schemes for the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential.We need to notice that it is very challenging to verify the unconditional energy stability for the time fractional Cahn-Hilliard and Allen-Cahn model,which have not been reported before.The main difficulty is due to the fact that a lot of extra interference terms would arise by introducing the time fractional operator.By reconsidering the properties of coefficients in the time fractional numerical dif-ferentiation formula,we overcome this issue successfully.The stabilized method and scalar auxiliary variable(SAV)approach are applied to construct the uncon-ditional efficient schemes.Finally,various 2D and 3D numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes. |
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