On High Order Methods For Solving Parabolic Equation And Space Fractional Diffusion Equation

In this paper we mainly consider the high order finite difference method for solving parabolic equation of two-dimensional and space fractional diffusion equations of one-dimensional and two-dimensional. In the first chapter we construct the compact ADI scheme of parabolic equation. And we achieve the fourth order accuracy and second or-der accuracy with respect to space and time dimensions, respectively. Furthermore, we design the Richardson extrapolation approach to improve the accuracy order of both time and space to six order. In the second chapter, based on the WSGD operators in [33], we build the CWSGD operators to approximate the Riemann-Liouville fractional derivatives. And we apply them to construct the finite difference schemes for solving one-dimensional and two-dimensional space fractional diffusion equations. The third order accuracy of s-pace direction is obtained. we give the unconditional stability and the convergence with respect to the discrete L2norm of the schemes.