High Accuracy And Efficient Numerical Methods For Two-sided Fractional Diffusion Equations

Fractional order differential equations are generalizations of classical differential equations.Over the past two decades,fractional differential equations have been used in modeling turbulent flow,chaotic dynamics of classical conservative systems,groundwater contaminant transport,and applications in biology,physics,chemistry,and even finance.The main reason for their popularity is that they are equipped with the ability of capturing nonlocal phenomena and long range interactions,and can model phenomena exhibiting anomalous diffusion that cannot be modeled adequately and accurately by canonical second-order diffusion equations.The aim of this paper is to study highly accurate and efficient numerical methods for the two-sided space fractional order diffusion equation,one of particular interest among miscella-neous fractional order differential equations.As a powerful weapon in the scientific computing and applications,the finite difference method(FDM),the finite element method(FEM)and the spectral method(SM)are very essential numerical methods for solving the space fractional diffusion equations.Due to nonlocal definitions of fractional derivatives,however,the imple-mentation of numerical methods(FEM,FEM and SM)will too often incur excessive memory storage and expensive calculations.To overcome these hurdles,one of the effective approach is to develop the high accuracy numerical methods and use lesser grids or mode bases numbers for the purpose of reducing the storage and computational cost,which composes the focus of this paper.Also because of definitions of fractional derivatives including singular kernel,the solution of fractional differential equations inherits the weak singularities at the boundary,which leads to low accuracy.How to get the high accuracy approximation in this case is another issue to be addressed in this paper.Firstly,we derive a new fourth-order difference approximation for the space fractional derivatives by using the weighted average of the shifted Grunwald formulae combining the com-pact technique.We present and prove the properties of proposed fractional difference quotient operator.Then we apply the new approximation formula to solve the one-and two dimensional space fractional diffusion equations.By the energy method,we prove the proposed quasi-compact difference scheme to be unconditionally stable and convergent in L2 norm for both one-and two dimensional cases.We give several numerical examples to confirm the theoretical results.Secondly,we present an efficient algorithm by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with nonsmooth solution.We revisit two popular finite difference schemes,the weighted shifted Grunwald difference(WSGD)scheme and the fractional centered difference(FCD)scheme,and show the stability of the schemes in maximum norm.Based on the analysis of leading singularity of exact solution for the underlying problem,we demonstrate that,with the use of the proposed algorithm,the improved WSGD and FCD schemes can achieve higher accuracy than the original ones for nonsmooth solution.To further improve the accuracy for solving problems with small fractional order,we also develop an extended algorithm dealing with two-term singularities correction.We give several numerical examples to validate our theoretical prediction.We show that both accuracy and convergence rate of numerical solutions can be significantly improved by using the proposed algorithms.Thirdly,we study regularity and numerical methods for the two-sided fractional diffusion equation with lower order terms.We show that the regularity of the solution in weighted Sobolev spaces can be greatly improved compared to that in standard Sobolev spaces.With this regularity,we prove higher-order convergence of a spectral Galerkin method.We propose spectral Petrov-Galerkin methods and provide optimal error estimates for the Petrov-Galerkin method.Numerical results are presented to verify our predicted convergence orders.Last but not the least,we develop a Galerkin approach for two-sided fractional differential equations with variable coefficients.By the product rule,we transform the problem into an equivalent formulation which additionally introduces the fractional low-order term,and adopt the Galerkin formulation under weak ellipticity,albeit the coerciveness may not hold for large variable-coefficient problems.Next,we prove that the numerical solution is uniquely solvable with sufficiently small step size by Garding’s inequality,and present error estimate under the reasonable assumption for regularity of the solution.Finally,we provide several numerical examples to further illustrate the fidelity and accuracy of the proposed theoretical results.