78 research outputs found
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
Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation
The equation 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
Stochastic Galerkin method for cloud simulation. Part II: a fully random Navier-Stokes-cloud model
This paper is a continuation of the work presented in [Chertock et al., Math.
Cli. Weather Forecast. 5, 1 (2019), 65--106]. We study uncertainty propagation
in warm cloud dynamics of weakly compressible fluids. The mathematical model is
governed by a multiscale system of PDEs in which the macroscopic fluid dynamics
is described by a weakly compressible Navier-Stokes system and the microscopic
cloud dynamics is modeled by a convection-diffusion-reaction system. In order
to quantify uncertainties present in the system, we derive and implement a
generalized polynomial chaos stochastic Galerkin method. Unlike the first part
of this work, where we restricted our consideration to the partially stochastic
case in which the uncertainties were only present in the cloud physics
equations, we now study a fully random Navier-Stokes-cloud model in which we
include randomness in the macroscopic fluid dynamics as well. We conduct a
series of numerical experiments illustrating the accuracy and efficiency of the
developed approach
A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries
In this paper, we describe how to construct a nite-dierence shock-capturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domain, possibly with moving boundary. The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls. Although the motion of the boundary could be coupled with the fluid, all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow. The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domain ΩR>Ω. We identify inner and ghost points. The inner points are the grid points located inside, while the ghost points are the grid points that are outside but have at least one neighbor inside. The evolution equations for inner points data are obtained from the discretization of the governing equation, while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables (density, velocities and pressure). Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains. Several numerical experiments are conducted to illustrate the validity of the method. We demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems
Mixing effectiveness depends on the source-sink structure: Simulation results
The mixing effectiveness, i.e., the enhancement of molecular diffusion, of a
flow can be quantified in terms of the suppression of concentration variance of
a passive scalar sustained by steady sources and sinks. The mixing enhancement
defined this way is the ratio of the RMS fluctuations of the scalar mixed by
molecular diffusion alone to the (statistically steady-state) RMS fluctuations
of the scalar density in the presence of stirring. This measure of the
effectiveness of the stirring is naturally related to the enhancement factor of
the equivalent eddy diffusivity over molecular diffusion, and depends on the
Peclet number. It was recently noted that the maximum possible mixing
enhancement at a given Peclet number depends as well on the structure of the
sources and sinks. That is, the mixing efficiency, the effective diffusivity,
or the eddy diffusion of a flow generally depends on the sources and sinks of
whatever is being stirred. Here we present the results of particle-based
simulations quantitatively confirming the source-sink dependence of the mixing
enhancement as a function of Peclet number for a model flow.Comment: 13 pages, 9 figures, RevTex4 macro
Uncertainty quantification for kinetic models in socio-economic and life sciences
Kinetic equations play a major rule in modeling large systems of interacting
particles. Recently the legacy of classical kinetic theory found novel
applications in socio-economic and life sciences, where processes characterized
by large groups of agents exhibit spontaneous emergence of social structures.
Well-known examples are the formation of clusters in opinion dynamics, the
appearance of inequalities in wealth distributions, flocking and milling
behaviors in swarming models, synchronization phenomena in biological systems
and lane formation in pedestrian traffic. The construction of kinetic models
describing the above processes, however, has to face the difficulty of the lack
of fundamental principles since physical forces are replaced by empirical
social forces. These empirical forces are typically constructed with the aim to
reproduce qualitatively the observed system behaviors, like the emergence of
social structures, and are at best known in terms of statistical information of
the modeling parameters. For this reason the presence of random inputs
characterizing the parameters uncertainty should be considered as an essential
feature in the modeling process. In this survey we introduce several examples
of such kinetic models, that are mathematically described by nonlinear Vlasov
and Fokker--Planck equations, and present different numerical approaches for
uncertainty quantification which preserve the main features of the kinetic
solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic
Equations
Non-local kinetic and macroscopic models for self-organised animal aggregations
The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient Ï” is varied from Ï”=1 (for 1D transport models) to Ï”=0 (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit Ï”â0, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams
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