Numerical Simulation For A Kind Of Fractional Diffusion Equations

Fractional-order difusion equations describe such phenomena as the signalingof biological cells, anomalous electronic difusion in nerve cells, viscoelastic andviscoplastic fow, and solute transport in groundwater, which is known as anomalousdifusion whose fux were found to be non-Fickian. Numerous experimental resultsshow that fractional-order difusion equations provide an adequate and accuratedescription of difuse processes that exhibit anomalous difusion, which can not bemodeled properly by second-order difusion equations. Therefore, it has become ahot spot to establish the theory of fractional-order difusion equations and developefciently numerical methods in the felds of computational mathematics and appliedmathematics.Similar to the second-order difusion equations, only few special fractional-order difusion problems can be found their analytic solutions by using analyticmethods such as Fourier transform, Laplace transform. One has to resort to numer-ical solutions by developing appropriate numerical procedures. Due to the non-localproperty of fractional diferential operators, their numerical methods often generatenon-sparse coefcient matrices with complicated structures. It requires computa-tional work of O(N3) and memory of O(N2) if the traditional Gaussian eliminationis used. The huge amount of computational work and memory makes the simulationprohibitively expensive when N is large enough. It is a challenge to develop fast numerical algorithm to approximate the fractional difusion equations.In this paper, we hope to design two kinds of fast algorithms. One is analternating direct implicit method, by combining the stability of implicit diferencemethod and the fast computation of alternating direct technique. The other is afast discontinuous Galerkin procedure base on the fast Fourier transform and thediscontinuous Galerkin method.1. The fast procedure for the time-fractional difusion equation in two-dimensionalspace. Based on the defnition of Caputo’s fractional derivative, the Caputo’s frac-tional derivative is approximated by a quadrature formula for the backward Eulerdiference scheme of the frst-order time derivative, and the Laplacian operator isdiscreted by the alternating direct implicit diference scheme. Thus, we set up analternating direct implicit method to solve time-fractional difusion equations intwo-dimensions. Numerical analysis and experiments show that the numerical solu-tion possesses unconditional stability and an optimal convergence rate. Its spatialcomputational work reduce to O(N2) from O(N3) of the traditional method, thememory requirement is obviously reduced.2. The fast algorithm for homogeneous Dirichlet boundary-value problem ofspace-fractional difusion equations in one-dimension. By adopting the semi-groupproperty and adjoint property of Riemann-Liouville fractional derivative we developthe discontinuous Galerkin weak form and prove its equivalence with the space-fractional difusion equations in one space dimension. We also prove that the weakform satisfes the hypothesis of the Lax-Milgram theorem, thus the solvability of theweak form in fractional space H1β2is obtained. Further a discontinuous Galerkinfnite element scheme based on the weak formulation is proposed. Numerical anal-ysis implies that the discontinuous Galerkin fnite element solution has an optimalH1β2norm convergence rate O(h1+β2). Under the assumption on the regularity of the solution to the adjoint problem, we derive an optimal L2norm convergenceorder O(h2).To solve the fnite element equation, we delicately decompose the coefcientmatrix as the sum of a block-diagonal matrix, a block-Toeplitz matrix and a specialmatrix. The special matrix also can be transformed as a block-Toeplitz matrix. Bynoting that the computational work is O(N log N) and the memory requirement isO(N) when the fast Fourier transform is used to solve the matrix-vector multiplica-tion with a Toeplitz coefcient matrix, we apply the fast Fourier transform and theconjugate gradient method to solve the algebraic equation, and then obtain the fastdiscontinuous Galerkin fnite element algorithm with the O(N log N) computationalwork and the O(N) memory requirement. Numerical experiments show the utilityof the fast method.