Numerical Analysis For Some Fractional Anomalous Diffusion Equations

Anomalous diffusion is one of the most ubiquitous phenomena in nature, especially present in various complex systems. In order to describe anomalous diffusion more ac-curately, different tools are restored by different scholars. Over the past two decades, many researchers found that the fractional calculus is very suitable for describing this phenomenon. Many kinds of anomalous diffusion equations are established. It is difficult to find the analytical solutions of these equations which are closer to the real word. Hence, it is significant to find the numerical solutions of these equations.The main objective of this thesis is to investigate the numerical solutions of three kinds of anomalous equations, namely, time fractional (radial) diffusion equations, frac-tional Klein-Kramers equations and fractional conservation laws. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our pro-posed numerical schemes are also established. And the numerical simulations are given to reveal the kinetics behaviors of these equations. The main work of this thesis contains the following three parts.In the first part, we consider the numerical solutions of two kinds of time fractional diffusion equations which are used to describe subdiffusion. We first develop two kinds of high order, easy for programming, unconditionally stable orthogonal spline collocation schemes to solve the time fractional diffusion equation. Then, we establish two kinds of implicit finite difference schemes for the time fractional radial diffusion equation. Applying the method of mathematical induction and discrete maximum principle, we prove that the given schemes are all unconditionally stable. Some numerical results and physical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes.In the second part, we focus on the numerical solutions of fractional Klein-Kramers equations. Firstly, we present the finite difference methods for numerically solving the time fractional Klein-Kramers equation and do the detailed stability and error analysis. The numerical examples are provided to confirm the theoretical results. Furthermore, we discuss the numerical solutions of Levy fractional Klein-Kramers dynamics. Explicit and implicit finite difference schemes are established for this equation. By introducing a gener-alized discrete maximum principle, we carefully check the detailed numerical stability and convergence of the numerical schemes. Some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. Numerical results confirm the effectiveness of our numerical schemes.In the third part, we discuss the numerical algorithms for fractional conservation laws. We present a semi-discrete Fourier spectral method for a periodic fractional conservation law with smooth solutions. The error estimation of the space semi-discrete scheme is rigorously established. And the fourth-order integrating factor-Runge-Kutta method is used to solve the semi-discrete system. The numerical results further confirm the spectral accuracy in space and fourth-order convergence in time. For the non-smooth case, we design fractional step method to deal with it. The numerical examples show that the proposed methods are effective for both smooth and discontinuous initial values'cases.