Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

Abstract

In this papers, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of the differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we also apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems. û n+ j+1 J = F (t n+ j J ,û n+ j J , ∆T J ), j = 0, 1, . . . , J − 1, u n = u (k) 0 .

On Parareal Algorithms for Semilinear Parabolic Stochastic PDEs

“…In particular, [1] deals with parabolic PDEs, and studies the stability and convergence properties, which may require regularity properties, depending on the choice of integrators. The application of the parareal algorithm for stochastic systems has been considered first in [2], and more recently in [11] for stochastic Schrödinger PDEs and in [24] for a class of stochastic differential equations. More precisely, in [11], parareal algorithms for stochastic Schrödinger equation with damping are studied with F being the exact solver and G being the exponential-θ scheme.…”