13 research outputs found

    Author Index Volume 231 (2009)

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    High-order Compact Difference Schemes for the Modified Anomalous Subdiffusion Equation

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    In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference operator. We apply these methods to fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by Fourier method in details. The convergence orders of these numerical schemes are O(τ2+h6)\mathcal {O}(\tau^2+h^6) and O(τ2+h8)\mathcal {O}(\tau^2+h^8), respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.Comment:

    Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations

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    We consider an initial-boundary value problem for tutα2u=f(t)\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t), that is, for a fractional diffusion (1<α<0-1<\alpha<0) or wave (0<α<10<\alpha<1) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near t=0t=0, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial L2L_2-norm, is of order k2+α+h2(k)k^{2+\alpha_-}+h^2\ell(k), uniformly in tt, where kk is the maximum time step, hh is the maximum diameter of the spatial finite elements, α=min(α,0)0\alpha_-=\min(\alpha,0)\le0 and (k)=max(1,logk)\ell(k)=\max(1,|\log k|). Here, we generalize a known result for the classical heat equation (i.e., the case α=0\alpha=0) by showing that at each time level tnt_n the solution is superconvergent with respect to kk: the error is of order (k3+2α+h2)(k)(k^{3+2\alpha_-}+h^2)\ell(k). Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation for any tt. Numerical experiments indicate that our theoretical error bound is pessimistic if α<0\alpha<0. Ignoring logarithmic factors, we observe that the error in the DG solution at t=tnt=t_n, and after postprocessing at all tt, is of order k3+α+h2k^{3+\alpha_-}+h^2.Comment: 24 pages, 2 figure
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