Fast Algorithm Of Subdiffusion Equations And Its Theoretical Analysis

Due to fractional derivatives include historical information,the storage and computation required by direct calculation with traditional methods are O(MN)and O(MN2),respectively,here M is the total number of spacial points and N is the total number of temporal points.In long time simulations,both the storage and computation are huge.In order to overcome these difficulties,we get the fast algorithms through approximating the kernel of Caputo derivative with the sum of exponentials.Applying the fast algorithms to the sub-diffusion problems,the needed storage and computation are reduced to O(MNexp)and O(MNNexp),respectively,where Nexp is the number of exponentials.In long time simulations,Nexp?N,thus the new numer-ical schemes improve the efficiency greatly.On the other hand,for the reaction sub-diffusion equations and nonlinear sub-diffusion equations,the solutions not only are singular at initial time but also could increase quickly at some time point far away from initial time.In order to get the numerical best temporal convergence order,it usually needs to calculate with general nonuniform temporal mesh,which brings huge difficulty to theoretical analysis.For reaction sub-diffusion problem,applying the discrete fractional Gronwall inequality,we prove that the new numerical schemes with general nonuniform time mesh are unconditional stable.Through estimating the global consistency errors,we analyze the convergence of the numerical schemes.For nonlinear sub-diffusion problem,we linearize the nonlinear term by Newton method and then establish a high efficient scheme.With the help of the discrete fractional Gronwall inequal-ity,a global consistency analysis and a discrete H2 energy method,we prove that the scheme is also unconditional stable and convergent.