Spectral And Discontinuous Galerkin Methods For Fractional Differential Equations

Abstract:Fractional differential equations have been used in numerical simulation of many en-gineering problems due to its global and memory properties. And the development of application of fractional calculus has made the solution of fractional differential equations an emergent task to solve. As constructed in local element and implemented independently, multi-domain spec-tral method and discontinuous Galerkin method have been confirmed to be numerical methods of high accuracy and adaptivity, which are suitable for problems defined on complex geometric domain. In recent years, these two methods have worked well for classical differential equations. In this thesis, based on multi-domain spectral method and discontinuous Galerkin method, we aim to construct and develop some high accurate and flexible methods for solution of fraction-al differential equations, which are suitable for solution of fractional differential equations on non-uniform mesh. Main results of this thesis are listed as follows,(1) With discontinuous Galerkin method is used for space discretization and finite difference method is used for time discretization, we constructed a finite difference-discontinuous Galerkin method for time fractional diffusion equation. L2stability and optimal convergence property of space semi-discretization are proved. Theoretical results are verified by numerical examples.(2) Based on the idea of multi-domain spectral method, in order to achieve high accura-cy and low cost, we constructed a hybrid approximation method for computing memory term of fractional derivatives, and then a multi-domain spectral method is constructed base on this. Considering the’short memory principle’of fractional derivative, we truncate the small long tail terms and transform the infinite step method to a fix number steps method. Stability region of the method is presented through analyzing eigenvalues of transfer matrices. The convergence of the method is confirmed through numerical examples. Numerical examples show that, the method is of N+1-α-th order convergence. This happened to remedy the weekness of finite difference-discontinuous Galerkin method, which is of first order accuracy in time and has limitation on the value of a.(3) Polynomial approximation is constructed for fractional derivative in single domain, and piece-wise polynomial approximation is constructed for fractional derivative in multi-domain. Error analysis of the constructed method is carried out based on properties of orthogonal poly-nomials. It is proved that, when N-[α] is odd, the proposed method converge with order N+1-[α]; when N-[α] is even, the proposed method converge with order N+1-α. By adding penalty terms to the equation, penalty spectral methods are constructed for space fractional advection equation and space fractional diffusion equation separately. Stability of the methods are analyzed through eigenvalue analysis and convergence of the method is verified through numerical examples.(4) Discontinuous Galerkin method is studied for fractional convection-diffusion equation with fractional Lalacian. To guarantee consistence and high accuracy of the method, we rewrite the fractional Laplacian as a composite of two first order derivative and2-α-th order fractional integral. Then low order system is obtained through introducing two auxiliary variables. Lo-cal discontinuous Galerkin method is introduced for space discretization of the system. With alternating direction fluxes are used for diffusion terms and monotone flux is used for nonlinear term, we derived local discontinuous Galerkin weak formulation and proved L2stability of space discretization. For fractional diffusion problem, we proved the method to be optimal convergent. For the fractional convection-diffusion, we proved the method to converge with order N+1/2. Stability and convergence order are verified through numerical examples.(5) Finite difference approximation is derived for a kind of generalized Caputo derivative. Collocation method is introduced for space discretization. Then a finite difference-collocation method is proposed for two-dimensional time fractional diffusion equation with generalized Ca-puto derivative. Stability of the scheme is proved and convergence order of the scheme is ana-lyzed through numerical examples.