Finite Difference Method Of(Tempered)Fractional Diffusion Equations

The continuous time random walk(CTRW) model, composed of waiting times and jump lengths, is a pillar of statistical physics to characterize the anomalous dynamics.The power-law waiting time distribution is generally used to describe the subdiffusion;and the power-law jump length distribution is applied to Lévy flights. Based on the corresponding CTRW models, the time, space, or time-space fractional diffusion equations are derived to govern the probability density function(PDF) of the particles. For the CTRW with the distribution of the tempered jump length |x|-(1+α)e-λ|x|, the corresponding PDF of the particles satisfies the tempered space fractional diffusion equation.The main research contents of the study are as follows:Firstly, we design semi-implicit schemes for the scalar time fractional reactiondiffusion equations. We theoretically prove that the numerical schemes are stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergence orders are 1 in time and 2 in space. As a concrete model, the subdiffusive predator-prey systems are discussed in detail. We use the provided numerical schemes to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical schemes preserve the positivity and boundedness.Secondly, the semi-implicit schemes for the nonlinear predator-prey reaction-diffusion model with the space-time fractional derivatives are discussed, where the space fractional derivatives are discretized by the fractional centered difference and shifted GrünwaldLetnikov algorithm. The stability and convergence of the semi-implicit schemes are analyzed in the L2 norm. We theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space. Then we show that the numerical solutions preserve the positivity and boundedness of analytical solutions. The presented numerical examples confirm the theoretical results and convergence orders.Thirdly, we provide the basic strategy of deriving the high order quasi-compact discretizations for Riemann-Liouville fractional derivative and tempered space fractional derivative. The CTRW underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution,its corresponding Fokker-Planck equation has space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more ’physical’ and the tempered space fractional diffusion equation appears.The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore,the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.Fourthly, we focus on providing the quasi-compact schemes for the tempered fractional diffusion equations. Not only its derivation but the proof of its numerical stability and convergence are different from the ones of the fractional diffusion equations. The detailed theoretical results are presented, and some techniques are introduced in the analysis: by using the generation function of the matrix and Weyl’s theorem, the stability and convergence of the derived schemes are strictly proved. Extensive numerical simulations are performed to show the effectiveness of the schemes, and the third order convergence is confirmed.