caa a22 4500 605497028 CHVBK 20210128100539.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10543-014-0531-z doi (NATIONALLICENCE)springer-10.1007/s10543-014-0531-z Local discontinuous Galerkin methods for fractional ordinary differential equations [Elektronische Daten] [Weihua Deng, Jan Hesthaven] 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 $$k+1$$ k + 1 in the $$L^2$$ L 2 norm and superconvergence of order $$k+1+\min \{k,\alpha \}$$ k + 1 + min { k , α } at the downwind point of each element. Here $$k$$ k is the degree of the approximation polynomial used in an element and $$\alpha $$ α ( $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] ) represents the order of the one-term FODEs. A generalization of this includes problems with classic $$m$$ m 'th-term FODEs, yielding superconvergence order at downwind point as $$k+1+\min \{k,\max \{\alpha ,m\}\}$$ 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. Springer Science+Business Media Dordrecht, 2014 Fractional ordinary differential equation nationallicence Local discontinuous Galerkin methods nationallicence Downwind points nationallicence Superconvergence nationallicence Generalized Mittag-Leffler function nationallicence Deng Weihua School of Mathematics and Statistics, Lanzhou University, 730000, Lanzhou, People's Republic of China aut Hesthaven Jan EPFL-SB-MATHICSE-MCSS, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland aut BIT Numerical Mathematics Springer Netherlands 55/4(2015-12-01), 967-985 0006-3835 55:4<967 2015 55 10543 https://doi.org/10.1007/s10543-014-0531-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10543-014-0531-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Deng Weihua School of Mathematics and Statistics, Lanzhou University, 730000, Lanzhou, People's Republic of China aut NATIONALLICENCE 700 1- Hesthaven Jan EPFL-SB-MATHICSE-MCSS, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland aut NATIONALLICENCE 773 0- BIT Numerical Mathematics Springer Netherlands 55/4(2015-12-01), 967-985 0006-3835 55:4<967 2015 55 10543