Investigation of moment models for population balance equations and radiative transfer equations

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This thesis explores moment approximations for population balance and radiative transfer equations, focusing on laser-induced thermotherapy for liver tumors. Moments are created by averaging over a variable against basis functions to generate spectral approximations. In the context of the population balance equation, three closures are examined: minimum entropy, polynomial, and Kershaw closures. The polynomial closure is purely spectral, but it cannot guarantee the realizability of moments, meaning the underlying distribution function may not remain non-negative. The minimum entropy method addresses this issue using a non-negative ansatz, though it incurs higher computational costs due to the need for an optimization algorithm to reconstruct the distribution function. The Kershaw closure, which preserves realizability, employs a linear combination of Dirac-delta distributions informed by realizability boundaries. The study also considers full and partial linear moment models alongside the diffusion approximation. To enhance the accuracy of the steady-state radiative transfer equation, an iterative solution scheme is accelerated using a multigrid approach based on Lebedev quadrature rules. Additionally, a more relevant model is proposed using the incompressible Navier-Stokes equation to simulate blood and cooling fluid flow, addressing the heterogeneous nature of the heat and radiative transfer equations. The Runge-Kutta discon