Convergence of Parareal Algorithms for PDEs with Fractional Laplacian and a Non-Constant Coefficient

Abstract

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.