Finite Difference Schemes For Solving Space-Fractional Diffusion Equations And Convection-Diffusion Equations

In recent decades, Fractional partial differential equations (FPDEs) have been more and more widely used in natural science and social science. Since frac-tional derivative can be used to describe the anomalous diffusion phenomenon, we apply fractional kinetics equations to model some complex systems which contain a number of anomalous diffusion phenomena. It’s difficult to get the an-alytical solutions of FPDEs, so it’s essential to research the numerical methods for solving FPDEs.There are some difficulties to obtain the FPDEs’numerical solutions. First-ly, fractional differential operators are nonlocal, which leads to the uncondition-ally unstable for numerical discretization to solve FPDEs. Secondly, Numerical methods for FPDEs tend to generate full coefficient matrices, which require com-putational cost of O(N3) and storage of O(N2), where N is the number of the grid point. Thirdly, the theoretical analysis for the numerical methods to solve FPDEs are more difficult. It’s essential to construct numerical methods for solv-ing FPDEs, which not only are stable, but also can save the computational cost and storage, in addition can do the theoretically analysis easily. As the one of the classical theories for integer order differential equations, the maximum principle provide the possibilities for presenting such methods. On the one hand it insures the stability for the numerical methods, on the other hand we can use the max-imum principle to do some priori estimates. In addition, the stability and the convergence are proved by the maximum principle. On this basis, combined with some classical numerical methods for solving the integer order differential equa-tions, we can construct some stable numerical methods for solving FPDEs, which not only save the computational cost and storage, but also do some theoretical analysis conveniently.In this thesis, based on the maximum principle and the definition of the R-L fractional derivative, we construct a second order operator to discrete the fractional derivative. In addition combined with the implicit Euler scheme and the Saul’ev scheme to discrete the time derivative, we present several finite dif-ference schemes for solving the space fractional diffusion equation and the space fractional diffusion-convection equation and prove the schemes’ stability and con-vergence. The thesis contains the following innovations and contents.Firstly, the background, questions, purpose, significance and study of the current situation at home and abroad for numerical solutions to solve FPDEs are introduced. In addition, some background information are given, such as several commonly used fractional derivative definitions, their equivalent relationship, their properties, and the basic knowledge of the maximum principle.Secondly, the second order operator satisfying the maximum principle is presented to approximate the fractional derivative. Combined with the implicit Euler method, some second order finite difference schemes for solving one-sided or two-sided space-fractional diffusion equation and space-fractional convection-diffusion equation are presented. In addition the stability and convergence of the schemes are proved using the maximum principle or the discrete energy method.Thirdly, combined the presented second order operator with the Saul’ev scheme, some kind of symmetric semi-implicit schemes are constructed for solving one-dimensional space-fractional diffusion equation and space-fractional convection-diffusion equation. In form, the presented finite difference scheme is implicit, but we can get the discrete solution of the scheme explicitly, so the computation and storage are saved greatly. In addition the uniform stability and the error estimate between the discrete solution and analytical solution theoretically performed. We theoretically prove and numerically verify that the error between the discrete and the analytical solution in discrete I2 norm is of order C(△t2h-2(1-α)+△t+h2), where a is the order of the space fractional derivative and At and h are the time and the space meshsizes.Fourthly, the second order operator defined on the non-integer node to ap-proximate the fractional derivative is given. Based on the maximum principle, a second order central finite difference scheme in space are presented for solving one-dimensional space-fractional diffusion equation, and it’s stability and con-vergence are proved using the maximum principle.