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The asymptotic stable region and long-time decay rate of solutions to linearhomogeneous Caputo time fractional ordinary differential equations (F-ODEs) areknown to be completely determined by the eigenvalues of the coefficient matrix.Very different from the exponential decay of solutions to classical ODEs,solutions of F-ODEs decay only polynomially, leading to the so-calledMittag-Leffler stability, which was already extended to semi-linear F-ODEs withsmall perturbations. This work is mainly devoted to the qualitative analysis ofthe long-time behavior of numerical solutions. By applying the singularityanalysis of generating functions developed by Flajolet and Odlyzko (SIAM J.Disc. Math. 3 (1990), 216-240), we are able to prove that both $\mathcal{L}$1scheme and strong $A$-stable fractional linear multistep methods (F-LMMs) canpreserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEsexactly as in the continuous case. Through an improved estimate of the discretefractional resolvent operator, we show that strong $A$-stable F-LMMs are alsoMittag-Leffler stable for semi-linear F-ODEs under small perturbations. For thenumerical schemes based on $\alpha$-difference approximation to Caputoderivative, we establish the Mittag-Leffler stability for semi-linear problemsby making use of properties of the Poisson transformation and the decay rate ofthe continuous fractional resolvent operator. Numerical experiments arepresented for several typical time fractional evolutional equations, includingtime fractional sub-diffusion equations, fractional linear system andsemi-linear F-ODEs. All the numerical results exhibit the typical long-timepolynomial decay rate, which is fully consistent with our theoreticalpredictions.

Mittag--Leffler stability of numerical solutions to time fractional ODEs