The High-order Numerical Algorithms For The Time Fractional Partial Differential Equations

This work is devoted to constructing high-accuracy numerical methods for the time fractional diffusion equations and establishing the corresponding error estimates.Firstly, we construct a high-order numerical algorithm for the one-dimensional time fraction-al sub-diffusion equation and establish the corresponding prior estimate. The method follows the idea of the weighted and shifted Griinwald-Letnikov difference operators [48,64]. Choosing shifts (p,q,r) =(0,-1,-2) and utilizing the equivalence of Riemann-Liouville derivative and Caputo derivative under some regularity assumptions, we derive a third-order accuracy formula to approx-imate Caputo fractional derivative. A new compact difference scheme (called the GL3 scheme) is derived for the fractional sub-diffusion equation by combining the average operator technique for the spatial discretization. The unconditional stability and convergence of the GL3 scheme are rigorously proved by the discrete energy method.Secondly, we discuss a high-accuracy finite difference scheme for the two-dimensional time frac-tional sub-diffusion equation. Following the techniques of constructing difference scheme in one-dimension, a high-order difference scheme both in time and in space is derived for the two-dimensional problem. It is proved that the difference scheme is unconditionally stable and convergent in L1 (L∞)-norm. There are essential differences on theoretical analysis techniques between the one-dimensional problem and the two-dimensional problem, since H1-norm can not be embedded into L∞-norm in two-dimension. Meanwhile, by adding a small item, we present a compact alternating direction im-plicit finite difference scheme (called compact ADI scheme) for the two-dimensional problem and demonstrate the convergence order and effectiveness of the compact ADI scheme with numerical examples.Thirdly, we consider an effective difference scheme for the time fractional diffusion-wave equation. Performing the Riemann-Liouville fractional integral operator on both sides of the equation, the problem is equivalently reformulated to the integral-differential equation. Based on the piecewise linear interpolation method, we derive a new discretization formula to approximate Riemann-Liouville fractional integral operator with second-order accuracy. And a difference scheme for the equivalent equation is presented. We present some numerical examples to numerically prove the convergence order and validity of the scheme.Finally, we focus on numerical algorithms with high spatial accuracy for the fourth-order fraction-al sub-diffusion equation with the first Dirichlet boundary conditions. We take linear combinations of the function uxx(x,t) at grid points x0,x1,x2,x3 and at grid points xM,xM-1,xM-2,xM-3, re-spectively, to deal with the first Dirichlet boundary conditions with fourth-order accuracy. Applying the L1 formula to approximate the temporal Caputo fractional derivative and combining the compact scheme for the spatial fourth-order derivative, a difference scheme is presented, which has fourth-order spatial accuracy uniformly. The unconditional stability and convergence of the presented difference scheme are analyzed by utilizing mathematical induction and discrete energy method.