Numerical simulation of a micro–macro model of concentrated suspensions

Abstract

SUMMARYWe deal here with the numerical simulation of a multiscale model of concentrated suspensions. We compare the deterministic solution procedure for the Fokker Planck equation with the Monte Carlo simulation of the stochastic di erential equation. In particular, we examine questions of variance reduction.

Macroscopic Limit of a One-Dimensional Model for Aging Fluids

“…In all our tests, the reference equation, namely (1.1) (or more precisely (6.2)), is Equation (8.1a) is solved explicitly, pointwise for each σ = 0, while the equation for σ = 0 is indeed solved using the conservation of the total mass of the density p. In short, the value of the density at zero is adjusted so that the integral of p is one. See [9] for more details. The advection equation (8.1b) is solved using an upwind finite difference scheme.…”

“…Since in theory it is posed on the whole real line, we need to truncate the domain and thus actually solve the equation on the bounded interval σ ∈ [−M σ , M σ ] (with periodic boundary conditions), for M σ = 10, with a constant space step ∆σ = Mσ Equation (8.1a) is solved explicitly, pointwise for each σ = 0, while the equation for σ = 0 is indeed solved using the conservation of the total mass of the density p. In short, the value of the density at zero is adjusted so that the integral of p is one. See [9] for more details. The advection equation (8.1b) is solved using an upwind finite difference scheme.…”

Well-Posedness of a Multiscale Model for Concentrated Suspensions

“…In fact, some numerical simulations performed by one of us in [3] show that even when that assumption is not satisfied at initial time t = 0, it is indeed satisfied for t > 0 arbitrarily small. We are unfortunately not able to establish this fact rigorously, but the numerical evidence mentioned above heuristically shows that condition (2.4) can be considered to be always satisfied, up to a change in the choice of the origin of times.…”