Semi-Discretized Stochastic Fiber Dynamics: Non-Linear Drag Force

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

“…Hence, γ(r, ṽ, h) = κ(r, ṽ, h) for all (r, ṽ, h) ∈ V 1 . We obtain that the Lagrange multiplier κ(r, v, h) 24), and the definition of A(r, v, h, γ) ∈ R m×m in (3.23). Thus, replacing the Lagrange remainder terms appearing due to Taylor expansions on the right hand side of (3.34) by their corresponding integral representations, it follows that λre (r, v, h) ∈ R m and Λre (r, v, h) ∈ R m×ℓ depend continuously on (r, v, h) ∈ D as well.…”

“…. , N , in three-dimensional Euclidean space, described by an SDAE of the form dr(t) = v(t) dt dv(t) = f int (r(t)) + f grav + f air r(t), v(t) dt + L air r(t), r(t) D turb r(t), v(t) dw(t) + ∇g(r(t)) dµ(t) g(r(t)) = 0, see [23,24]. Here r(t) = (r i (t)) i∈{1,...,N } is the 3N -dimensional position vector, v(t) = (v i (t)) i∈{1,...,N } is the corresponding 3N -dimensional velocity vector and w is a N -dimensional Brownian motion.…”

“…The process µ = (µ(t)) t 0 is an m-dimensional semimartingale, implicitly given as the Lagrange multiplier to the holonomic constraint g(r(t)) = 0. It thus determines the constraint force, which, according to d'Alembert's principle, acts orthogonal to the tangent space T x M = {y ∈ R n : Dg(x)y = 0} at the current position x ∈ M. Constrained SDEs of the form (1.1) occur in various applications, ranging from molecular dynamics [19-22, 40, 41] to models for the dynamics of fibers in turbulent airflows in the context industrial production processes of non-woven textiles [23,24].…”

“…We assume that a and B are locally Lipschitz continuous, of polynomial and linear growth, respectively, and that the mapping (x, y) → (y, a(x, y)) satisfies a one-sided linear growth condition. Note that the diffusion coefficient B is allowed to depend on both position and velocity, which is motivated by [23,24]. The derivatives of g are assumed to satisfy suitable boundedness and non-degeneracy conditions in a neighborhood of the manifold.…”