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Author: Patterson, Robert I.A. × Subject: Coagulation ×

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Now showing 1 - 3 of 3
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    A stochastic weighted particle method for coagulation-advection problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Patterson, Robert I.A.; Wagner, Wolfgang
    A spatially resolved stochastic weighted particle method for inception--coagulation--advection problems is presented. Convergence to a deterministic limit is briefly studied. Numerical experiments are carried out for two problems with very different coagulation kernels. These tests show the method to be robust and confirm the convergence properties. The robustness of the weighted particle method is shown to contrast with two Direct Simulation Algorithms which develop instabilities.
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    Bilinear coagulation equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heydecker, Daniel; Patterson, Robert I.A.
    We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time tg ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.
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    Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Patterson, Robert I.A.
    Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of d-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, textitStoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.