Towards essential improvement for the Parareal-TR and Parareal-Gauss4 algorithms

Parareal is an iterative algorithm and is characterized by two propagators GG and FF, which are respectively associated with large step size ΔTΔT and small step size ΔtΔt, where ΔT=JΔtΔT=JΔt and J≥2J≥2 is an integer. For symmetric positive definite (SPD) system u′(t)+Au(t)=g(t) arising from semi-discretizing time-dependent PDEs, if we fix the GG-propagator to the Backward-Euler method and choose for FF some LL-stable time-integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy ρ≈13, ∀J≥2∀J≥2 and ∀σ(A)⊂[0,+∞)∀σ(A)⊂[0,+∞), where σ(A)σ(A) is the spectrum of the matrix AA. However, this result does not hold when time-integrators that lack LL-stability, such as the Trapezoidal rule and the 4th-order Gauss RK method, are chosen as the FF-propagator. The parareal algorithms using these two methods for the FF-propagator are denoted by Parareal-TR and Parareal-Gauss4. In this paper, we propose a strategy to let these two parareal algorithms possess such a uniform convergence property. The idea is to choose an LL-stable propagator F˜ and on each coarse time-interval [Tn,Tn+1][Tn,Tn+1] we perform first two steps of F˜, then followed by J−2J−2 steps of FF. Precisely, for the Trapezoidal rule we select the 2nd-order SDIRK method as the F˜-propagator, and for the 4th-order Gauss RK method we select the 4th-order Lobatto III-C method as the F˜-propagator. Numerical results are given to support our theoretical conclusions.