Variational computational methods for multiscale kinetic equations and related problems

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Variational computational methods for multiscale kinetic equations and related problems

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2022-07

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The Knudsen number $\varepsilon$ plays important role in kinetic equations, whose magnitudecan vary dramatically according to different physics phenomena, therefore introducing stiffness to the kinetic system and bringing difficulties in designing accurate and efficient numerical methods. One standard strategy to overcome this issue is to design the asymptotics preserving (AP) scheme, which aims at a unified solver that captures the asymptotic limit at the discrete level. In this thesis, we focus on developing AP schemes through variational approaches for multi-scale kinetic equations and their applications to the related problems. The first system we study is the Vlasov-Poisson-Fokker-Planck(VPFP) system with the high field scaling, for which we develop an AP scheme based on the well-known variational JKO scheme that captures the correct high-field limit. In addition, we develop an accelerated proximal quasi-Newton algorithm with partial second order information in computing the JKO scheme and further prove its super-linear convergence rate. The second equation is the granular kinetic equation with radially symmetric attractive collision kernel. The main feature of this equation is the tendency of clustering, which can possibly generate a blow-up solution. The spatially homogeneous equation is well studied, however, the behavior of the solution to the full kinetic equation is still a mystery. In particular, it is not yet known whether the concentration in velocity space would affect the regularity in position space. We intend to tackle this problem numerically and expect that this will shed some light on the global well-posedness of the equation. To do so, we leverage the JKO scheme with fisher information to produce physics relevant numerical solutions, along with a mesh refinement strategy that dynamically detects the potential blow-up region and then re-distributes more grid points to maintain high resolution within the potential blow-up region. We then make conjectures based on the numerical evidence. The third equation is the L\'evy-Fokker-Planck(LFP) equation with the fractional diffusive scaling. There are two main challenges in designing efficient and accurate schemes. One comes from a two-fold nonlocality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. To resolve this challenge, we use a change of variable to convert the unbounded domain into a bounded one and then apply the Chebyshev polynomial based pseudo-spectral method. The second challenge is to capture the transition from fractional Laplacian on velocity within the microscopic model to fractional Laplacian on position within the asymptotic limit system. To resolve this issue, we design an AP scheme based on a novel micro macro decomposition technique. The fourth equation is the steady radiative transfer equation (RTE) with diffusive scaling. To solve the equation, we leverage the physics-informed neural networks (PINNs) which seek a neural network approximator to the solution that minimizes the $L^2$ loss function governed by the underlying PDE equations. However, the vanilla PINNs fail to capture the correct diffusion limit and the boundary layer when an-isotropic boundary condition is considered. To capture the diffusion limit, we propose a new formulation of the loss based on the macro-micro decomposition and prove that the new loss function is uniformly stable with respect to $\eps$ in the sense that the $L^2$-error of the neural network solution is uniformly controlled by the loss. To resolve the boundary layer issue, we include a boundary layer corrector that carries over the sharp transition part of the solution and leaves the rest easy to be approximated.

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University of Minnesota Ph.D. dissertation. 2022. Major: Mathematics. Advisor: Li Wang. 1 computer file (PDF); 162 pages.

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xu, wuzhe. (2022). Variational computational methods for multiscale kinetic equations and related problems. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/241700.

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