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

摘要

Parareal is an iterative algorithm and is characterized by two propagators G and F, which are respectively associated with large step size ΔT and small step size Δt, where ΔT=JΔt and J≥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 G-propagator to the Backward-Euler method and choose for F some L-stable time-integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy ρ≈13, ∀J≥2 and ∀σ(A)⊂[0,+∞), where σ(A) is the spectrum of the matrix A. However, this result does not hold when time-integrators that lack L-stability, such as the Trapezoidal rule and the 4th-order Gauss RK method, are chosen as the F-propagator. The parareal algorithms using these two methods for the F-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 L-stable propagator F˜ and on each coarse time-interval [Tn,Tn+1] we perform first two steps of F˜, then followed by J−2 steps of F. 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.