TY - JOUR
T1 - Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 746
EP - 763
PY - 2018
DA - 2018/10
SN - 8
DO - http://doi.org/10.4208/eajam.220418.210718
UR - https://global-sci.org/intro/article_detail/eajam/12817.html
KW - Parareal method, time-varying problem, convergence analysis, parameter optimisation.
AB - <p style="text-align: justify;">The convergence of Parareal-Euler and -LIIIC2 algorithms using the backward
Euler method as a $\mathscr{G}$-propagator for the linear problem $U^′
(t)+α(t)A^ηU(t)=f(t)$ with
a non-constant coefficient $α$ is studied. We propose to employ the propagator $G$ to a
constant model $U^′(t)+βA^ ηU(t)=f(t)$ with a special coefficient β instead of applying
both propagators $\mathscr{G}$ and $\mathscr{F}$ to the same target model. We established a simple formula
to find an optimal parameter $β_{opt}$, minimising the convergence factor for all mesh ratios.
Numerical results confirm the proximity of theoretical optimal $β_{opt}$ to the optimal
numerical parameter.</p>