Analysis of a splitting scheme for a class of random nonlinear partial differential equations

Abstract

In this paper, we consider a Lie splitting scheme for a nonlinear partial differential equation driven by a random time-dependent dispersion coefficient. Our main result is a uniform estimate of the error of the scheme when the time step goes to 0. Moreover, we prove that the scheme satisfies an asymptotic-preserving property. As an application, we study the order of convergence of the scheme when the dispersion coefficient approximates a (multi)fractional process.

Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion

“…The time interval [0, T max ] of simulation is discretized considering t n = n∆t, n = 0, ..., N t where ∆t = Tmax Nt and N t is the number of time step. Equation (46) is then solved using the classical Strang splitting scheme (see for example [14,9,2]). This method can be summarized as follow:…”

Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion

“…In addition, order estimates have been established when the Brownian motion W in (1) is replaced by another process. For instance, if W is an α-Hölder function (α ∈ (0, 1)) then the order is α ( [9]), or if W is a fractional Brownian motion with Hurst index H ∈ (0, 1) then the order is H ( [8]). As in many other works dealing with splitting schemes for other deterministic or random equations (see for instance [10,4,1,2,8,9]), the study of the order crucially involves estimates of the local error.…”

Numerical analysis of the Gross-Pitaevskii Equation with a randomly varying potential in time

“…The proof uses a truncation on the nonlinearity. Moreover, in [15], the authors propose a Lie time-splitting scheme for a nonlinear partial differential equation driven by a random time-dependent dispersion coefficient. In this case the nonlinearity is supposed to be Lipschitz, but the dispersion coefficient can approximate a fractional Brownian motion.…”