Stochastic Analysis for Lagrangian Particles Simulation

Radu Maftei 1
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : This thesis broadly concerns colloidal particle simulation which plays an important role in understanding two-phase flows. More specifically, we track the particles inside a turbulent flow and model their dynamics as a stochastic process, their interactions as perfectly elastic collisions where the influence of the flow is modelled by a drift on the velocity term. By coupling each particle and considering their relative position and velocity, the perfectly elastic collision becomes a specular reflection condition. We put forward a time discretisation scheme for the resulting Lagrange system with specular boundary conditions and prove that the convergence rate of the weak error decreases at most linearly in the time discretisation step. The evidence is based on regularity results of the Feynman-Kac PDE and requires some regularity on the drift. We numerically experiment a series of conjectures, amongst which the weak error linearly decreasing for drifts that do not comply with the theorem conditions. We test the weak error convergence rate for a Richardson Romberg extrapolation and the effectiveness of the Multilevel Monte Carlo algorithm. We finally deal with Lagrangian/Brownian approximations by considering a Lagrangian system where the velocity component behaves as a fast process. We control the weak error between the position of the Lagrangian system and an appropriately chosen uniformly elliptic diffusion process and subsequently prove a similar control by introducing a specular reflecting boundary on the Lagrangian and an appropriate reflection on the elliptic diffusion.
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Radu Maftei. Stochastic Analysis for Lagrangian Particles Simulation. Probability [math.PR]. Université Côte D'Azur; Ecole Doctorale n°364 « Sciences Fondamentales et Appliquées », 2017. English. ⟨tel-01681293⟩

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