Numerical Study Of Fractional Flow Equations

Fractional differential equations have been attracted considerable attention in re-cent years, since they have applications in physics, geology, biology, chemistry, and even finance. As there are very few cases of fractional differential equations in which the explicit analytical solutions are available, numerical methods become major ways and then have been developed intensively. Under the assumption of seepage flow conti-nuity and the traditional Darcy's law, the traditional percolation equations have been applied successfully in groundwater hydraulics, groundwater dynamics and fluid dy-namics in porous media. A fractional Darcy's law with Riemann-Liouville fractional derivatives is proposed in 1998 for realistically describing the movement of solute in a non-homogeneous porous medium. Due to the reality that the seepage flow is neither continuous nor rigid, the fractional percolation equations are obtained from the tra-ditional percolation equations by replacing the integer space derivatives by fractional space derivatives.In Chapter 1, the history of the fractional calculus is reviewed. Some theoret-ical and numerical results in recent years of the fractional differential equations are presented. Besides, we introduce the physical background and new results for the fractional percolation equations.In Chapter 2, the Grunwald approximation of the fractional Riemann-Liouville derivative is introduced. A general approximation for the mixed fractional derivatives is established. Numerical experiments for the formula is given.In Chapter 3, the finite difference methods for the one-dimensional percolation equations are considered. An implicit back-Euler finite difference method and a Crank-Nicolson method based on the general approximation in Chapter 2 are studied. Con-sistency, stability and convergence of the methods are analyzed. An approach based on the Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. The resulting linear systems are Toeplitz-like and solved fast by the preconditioned conju-gate gradient normal residual method with a suitable circulant preconditioner which reduces the number of iterations to be mesh size independent. By the fast Fourier transform, the fast method only requires computational work of O(N log N) and stor-age of O(N) per time step. Numerical experiments are presented to verify the accuracy and efficiency of the fast method.In Chapter 4, we study the finite difference methods for the two-dimensional percolation equations. Based on the approximation proved in Chapter 1, the first-order and second-order finite difference method are developed. Consistency, stability and convergence of the methods are established. Thanks to the Toeplitz-like structure of the coefficient matrix of the finite difference method, a fast method which has a computational work of O(N log N) per iteration and a memory requirement of O(N) is proposed and tested in the numerical examples.In Chapter 5, we concern on the fractional percolation equation with Dirichlet and fractional Robin/Neumann boundary conditions. The analysis reflects that the constraint condition for the unconditioned stability is weaker than the case with pure Dirichlet boundary condition. We present a fast method for the resulting linear system with a proper preconditioner and give numerical experiments to illustrate the efficiency.We draw our conclusions and list our future consideration in Chapter 6.