This paper discusses the upwinded local discontinuous Galerkin methods for
the one-term/multi-term fractional ordinary differential equations (FODEs). The
natural upwind choice of the numerical fluxes for the initial value problem for
FODEs ensures stability of the methods. The solution can be computed element by
element with optimal order of convergence k+1 in the L2 norm and
superconvergence of order k+1+min{k,α} at the downwind point of each
element. Here k is the degree of the approximation polynomial used in an
element and α (α∈(0,1]) represents the order of the one-term
FODEs. A generalization of this includes problems with classic m'th-term
FODEs, yielding superconvergence order at downwind point as
k+1+min{k,max{α,m}}. The underlying mechanism of the
superconvergence is discussed and the analysis confirmed through examples,
including a discussion of how to use the scheme as an efficient way to evaluate
the generalized Mittag-Leffler function and solutions to more generalized
FODE's.Comment: 17 pages, 7 figure