OPTIMAL LONG-TIME DECAY RATE OF NUMERICAL SOLUTIONS FOR NONLINEAR TIME-FRACTIONAL EVOLUTIONARY EQUATIONS

The solution of the nonlinear initial-value problem D(t)(alpha)y(t)=-lambda y(t)(gamma) for t > 0 with y(0)>0, where D-t(alpha) is the Caputo derivative of order alpha is an element of (0,1) and lambda, gamma are positive parameters, is known to exhibit O(t(-alpha/gamma)) decay as t -> infinity. No corresponding result for any discretization of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving (CM-preserving) schemes (which includes the L1 and Grunwald-Letnikov schemes) on uniform meshes {t(n):=n(h)}(n=0)(infinity) the discrete solution also has O(t(n)(-alpha/gamma)) decay as t(n)->infinity. This result is then extended to CM-preserving discretizations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and p-Laplace equations. For the L1 scheme, the O(t(n)(-alpha/gamma)) decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.