An asymptotic preserving scheme for kinetic models with singular limit

We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.Comment: 24 pages, 6 figure

A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge

Convergence of a Particle Method and Global Weak Solutions for a Family of Evolutionary PDEs

Abstract: In this talk, I will present global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations, which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established

Integration of the EPDiff equation by particle methods

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold

A New Suburban Elysium: A Headstone for the Dying Periphery

The proposed “cemetery” and retail center for the Idora neighborhood of Youngstown, Ohio is the result of one research semester and one design semester. The design proposal arose from dissatisfaction with the architectural community’s propensity for using jargon and clichés when describing the contemporary suburban condition. Many critics and commentators understand suburbia through the lens of the postwar period. It has been suggested that suburbia was developed for use as a media weapon – and thus, at the conclusion of the Cold War, should have been rendered architecturally irrelevant. However, suburbia has remained stagnant. Design standards employed by developers continue to operate in support of an image-making regime. The image of domestic bliss suggests that Americans are capable of only one, homogenous form of existence. My thesis argues that a contemporary notion of suburbia can in fact be achieved by embracing its history and recognizing the overall ex-urban fabric as an occupiable historical document. A retail center – modelled on the strip mall typology – can behave as an antidote to suburban anxiety, when paired with a specific architectural language that establishes a ground-plane manipulation in which suburban homes can be recalled

The Gap-Tooth Method in Particle Simulations

We explore the gap-tooth method for multiscale modeling of systems represented by microscopic physics-based simulators, when coarse-grained evolution equations are not available in closed form. A biased random walk particle simulation, motivated by the viscous Burgers equation, serves as an example. We construct macro-to-micro (lifting) and micro-to-macro (restriction) operators, and drive the coarse time-evolution by particle simulations in appropriately coupled microdomains (teeth) separated by large spatial gaps. A macroscopically interpolative mechanism for communication between the teeth at the particle level is introduced. The results demonstrate the feasibility of a closure-on-demand approach to solving hydrodynamics problems

Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation

The equation $$\partial_tu=u\partial^2_{xx}u-(c-1)(\partial_xu)^2$$ is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock, therefore we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.Comment: 19 pages, includes 7 figure