The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations

The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations. Two applications are investigated, linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication-type equations for which it is unclear if the convergence analysis applies. The numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature. The convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions. Cutoff can also be used with other discretization methods such as collocation, finite volume, finite element, and spectral methods and for the computation of positive solutions.Comment: 19 pages, 41 figure

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes.Comment: 29 page

Analysis and Numerics of Stochastic Gradient Flows

In this thesis we study three stochastic partial differential equations (SPDE) that arise as stochastic gradient flows via the fluctuation-dissipation principle. For the first equation we establish a finer regularity statement based on a generalized Taylor expansion which is inspired by the theory of rough paths. The second equation is the thin-film equation with thermal noise which is a singular SPDE. In order to circumvent the issue of dealing with possible renormalization, we discretize the gradient flow structure of the deterministic thin-film equation. Choosing a specific discretization of the metric tensor, we resdiscover a well-known discretization of the thin-film equation introduced by Grün and Rumpf that satisfies a discrete entropy estimate. By proving a stochastic entropy estimate in this discrete setting, we obtain positivity of the scheme in the case of no-slip boundary conditions. Moreover, we analyze the associated rate functional and perform numerical experiments which suggest that the scheme converges. The third equation is the massive $\varphi^4_2$-model on the torus which is also a singular SPDE. In the spirit of Bakry and Émery, we obtain a gradient bound on the Markov semigroup. The proof relies on an $L^2$-estimate for the linearization of the equation. Due to the required renormalization, we use a stopping time argument in order to ensure stochastic integrability of the random constant in the estimate. A postprocessing of this estimate yields an even sharper gradient bound. As a corollary, for large enough mass, we establish a local spectral gap inequality which by ergodicity yields a spectral gap inequality for the $\varphi^4_2$- measure

NONLINEAR EVOLUTIONARY PDEs IN IMAGE PROCESSING AND COMPUTER VISION

Evolutionary PDE-based methods are widely used in image processing and computer vision. For many of these evolutionary PDEs, there is little or no theory on the existence and regularity of solutions, thus there is little or no understanding on how to implement them effectively to produce the desired effects. In this thesis work, we study one class of evolutionary PDEs which appear in the literature and are highly degenerate. The study of such second order parabolic PDEs has been carried out by using semi-group theory and maximum monotone operator in case that the initial value is in the space of functions of bounded variation. But the noisy initial image is usually not in this space, it is desirable to know the solution property under weaker assumption on initial image. Following the study of time dependent minimal surface problem, we study the existence and uniqueness of generalized solutions of a class of second order parabolic PDEs. Second order evolutionary PDE-based methods preserve edges very well but sometimes they have undesirable staircase effect. In order to overcome this drawback, fourth order evolutionary PDEs were proposed in the literature. Following the same approach, we study the existence and regularity of generalized solutions of one class of fourth order evolutionary PDEs in space of functions of bounded Hessian and bounded Laplacian. Finally, we study some evolutionary PDEs which do not satisfy the parabolicity condition by adding a regularization term. Through the rigorous study of evolutionary PDEs which appear in the literature of image processing and computer vision, we provide a solid theoretical foundation for them which helps us better understand the behaviors and properties of them. The existence and regularity theory is the first step toward effective numerical scheme. The regularity results also answer the questions to which function spaces the solutions of evolutionary PDEs belong and the questions if the processing results have the desired properties

The role of advection in phase-separating binary liquids

Using the advective Cahn-Hilliard equation as a model, we illuminate the role of advection in phase-separating binary liquids. The advecting velocity is either prescribed, or is determined by an evolution equation that accounts for the feedback of concentration gradients into the flow. Here, we focus on passive advection by a chaotic flow, and coupled Navier-Stokes Cahn-Hilliard flow in a thin geometry. Our approach is based on a combination of functional-analytic techniques, and numerical analysis. Additionally, we compare and contrast the Cahn-Hilliard equation with other models of aggregation; this leads us to investigate the orientational Holm-Putkaradze model. We demonstrate the emergence of singular solutions in this system, which we interpret as the formation of magnetic particles. Using elementary dynamical systems arguments, we classify the interactions of these particles.Comment: Ph.D. Thesis, Imperial College London, February 200

Analysis and applications of dynamic density functional theory

Classical fluid mechanics and, in particular, the general compressible Navier-Stokes-Fourier equations, have long been of great use in the prediction and understanding of the flow of fluids in various scenarios. While the classical theory is well established in increasingly rigorous mathematical frameworks, the atomistic properties and microscopic processes of fluids must be considered by other means. A central problem in fluid mechanics concerns capturing microscopic effects in meso/macroscopic continuum models. With more attention given to the non-Newtonian properties of many naturally occurring fluid flows, resolving the gaps between the atomistic viewpoint and the continuum approach of Navier-Stokes-Fourier is a rich and open field. This thesis centres on the modelling, analysis and computation of one continuum method designed to resolve the highly multiscale nature of non-equilibrium fluid flow on the particle scale: Dynamic Density Functional Theory (DDFT). A generalised version of DDFT is derived from first principles to include: driven flow, inertia and hydrodynamic interactions (HI) and it is observed that the equations reproduce known dynamics in heuristic overdamped and inviscid limits. Also included are rigorous, analytical derivations of the short-range lubrication forces on particles at low Reynolds number, with accompanying asymptotic theory, uniformly valid in the entire regime of particle distances and size ranges, which were previously unknown. As well as demonstrating an improvement on known classical results, these calculations were determined necessary to comply with the continuous nature of the integro-differential equations for DDFT. The numerical implementation of the driven, inertial equations with short range HI for a range of colloidal systems in confining geometries is also included by developing the pseudo-spectral collocation scheme 2DChebClass [67]. A further area of interest for non-equilibrium fluids is mathematical well–posedness. This thesis provides, for the first time, the existence and uniqueness of weak solutions to an overdamped DDFT with HI, as well as a rigorous investigation of its equilibrium behaviour