Finite Difference Methods For Fractional Convection Equations And The Related Problems

Due to the nonlocality of fractional calculus operators,fractional calculus has distinctive advantages in describing materials or processes with hereditary or memorial properties.Recently,the modeling and numerical computation about fractional differential equations have been widely applied in many fields of science and engineering.This thesis is dedicated to two aspects:constructing efficient numerical methods for nonlinear fractional differential equations,establishing fractional convection equations to model anomalous convection processes and developing numerical methods for these equations.Firstly,we introduce the fractional convection operator for the first time to describe anomalous convection phenomenon in the process of solute transport and present its discrete form with second order convergence.In order to get insight into the fractional convection behavior visually,a numerical scheme for the fractional convection-diffusion equation is constructed.Then we investigate the fractional convection dominated-diffusion and the fractional convection dominated-fractional diffusion equation.The interesting fractional convection phenomenons are displayed through numerical simulation.Secondly,we derive the fractional convection equations(FCEs)to model anomalous convection processes.Through using a continuous time random walk(CTRW)model with power-law jump length distributions,we formulate the FCEs depicted by Riesz derivatives with order in(0,1).The numerical methods for Riesz derivatives with order lying in(0,1)are constructed as well.Then the numerical approximations to FCEs are studied in details.By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendroff method in time,and the second order numerical method to Riesz derivative in space,we respectively obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for FCE with second order convergence both in time and in space.The accuracy and efficiency of the derived methods are verified by numerical tests.The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.Thirdly,we investigate coupled nonlinear fractional convection system to depict com-plex fluid dynamics.An implicit finite difference scheme is constructed to numerically approximate the derived system,which contributes to a nonlinear discrete system.In order to reduce computational cost,we use the extrapolation technique to linearize the original nonlinear system.The constructed method is conditionally stable and convergent with second order both in temporal and spatial directions,which is verified by numerical tests.We also provide numerical simulations to observe the evolution of coupled fractional convection process.Fourthly,finite difference methods with non-uniform meshes for solving nonlinear fractional differential equations are presented,where the non-equidistant stepsize is non-decreasing.The rectangle formula and trapezoid formula are proposed based on the non-uniform meshes.Combining the above two methods,we then establish the predictor-corrector scheme.The error and stability analysis are carefully investigated.At last,numerical examples are carried out to verify the theoretical analysis.Besides,the comparisons between numerical schemes performed on non-uniform and those on uniform meshes are given,where the non-uniform meshes show the better performance when dealing with the less smooth problems.At last,we propose an implicit-explicit scheme combining with the fast solver in space to solve two-dimensional nonlinear time-fractional subdiffusion equation.The applications of implicit-explicit scheme and fast solver will smartly enhance the computational efficien-cy.Due to the nonsmoothness(or low regularities)of solutions to fractional differential equations,correction terms are introduced in the proposed scheme to improve the accuracy of error.The stability and convergence of the present scheme are also investigated.Com-bining the nonuniform meshes,we make an extension of the presented implicit-explicit scheme.Numerical examples are carried out to demonstrate the efficiency and applicability of the derived scheme for both linear and nonlinear fractional subdiffusion equations with non-smooth solutions.Comparisons of different schemes including the implicit-explicit scheme,the implicit scheme and the scheme based on nonuniform meshes are also provided.