1,873 research outputs found
Scaling behavior in interacting systems: joint effect of anisotropy and compressibility
Motivated by the ubiquity of turbulent flows in realistic conditions, effects
of turbulent advection on two models of classical non-linear systems are
investigated. In particular, we analyze model A (according to the
Hohenberg-Halperin classification [1]) of a non-conserved order parameter and a
model of the direct bond percolation process. Having two paradigmatic
representatives of distinct stochastic dynamics, our aim is to elucidate to
what extent velocity fluctuations affect their scaling behavior. The main
emphasis is put on an interplay between anisotropy and compressibility of the
velocity flow on their respective scaling regimes. Velocity fluctuations are
generated by means of the Kraichnan rapid-change model, in which the anisotropy
is due to a distinguished spatial direction n and a correlator of the velocity
field obeys the Gaussian distribution law with prescribed statistical
properties. As the main theoretical tool, the field-theoretic perturbative
renormalization group is adopted. Actual calculations are performed in the
leading (one-loop) approximation. Having obtained infra-red stable asymptotic
regimes, we have found four possible candidates for macroscopically observable
behavior for each model. In contrast to the isotropic case, anisotropy brings
about enhancement of non-linearities and non-trivial regimes are proved to be
more stable
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
Computational modelling of iron-ore mineralisation with stratigraphic permeability anisotropy
This study develops a computational framework to model fluid transport in sedimentary basins, targeting iron ore deposit formation. It offers a simplified flow model, accounting for geological features and permeability anisotropy as driving factors. A new finite element method lessens computational effort, facilitating robust predictions and cost-effective exploration. This methodology, applicable to other mineral commodities, enhances understanding of genetic models, supporting the search for new mineral deposits amid the global energy transition
Lagrangian Numerical Methods for Ocean Biogeochemical Simulations
We propose two closely--related Lagrangian numerical methods for the
simulation of physical processes involving advection, reaction and diffusion.
The methods are intended to be used in settings where the flow is nearly
incompressible and the P\'eclet numbers are so high that resolving all the
scales of motion is unfeasible. This is commonplace in ocean flows. Our methods
consist in augmenting the method of characteristics, which is suitable for
advection--reaction problems, with couplings among nearby particles, producing
fluxes that mimic diffusion, or unresolved small-scale transport. The methods
conserve mass, obey the maximum principle, and allow to tune the strength of
the diffusive terms down to zero, while avoiding unwanted numerical dissipation
effects
Turing conditions for pattern forming systems on evolving manifolds
The study of pattern-forming instabilities in reaction-diffusion systems on
growing or otherwise time-dependent domains arises in a variety of settings,
including applications in developmental biology, spatial ecology, and
experimental chemistry. Analyzing such instabilities is complicated, as there
is a strong dependence of any spatially homogeneous base states on time, and
the resulting structure of the linearized perturbations used to determine the
onset of instability is inherently non-autonomous. We obtain general conditions
for the onset and structure of diffusion driven instabilities in
reaction-diffusion systems on domains which evolve in time, in terms of the
time-evolution of the Laplace-Beltrami spectrum for the domain and functions
which specify the domain evolution. Our results give sufficient conditions for
diffusive instabilities phrased in terms of differential inequalities which are
both versatile and straightforward to implement, despite the generality of the
studied problem. These conditions generalize a large number of results known in
the literature, such as the algebraic inequalities commonly used as a
sufficient criterion for the Turing instability on static domains, and
approximate asymptotic results valid for specific types of growth, or specific
domains. We demonstrate our general Turing conditions on a variety of domains
with different evolution laws, and in particular show how insight can be gained
even when the domain changes rapidly in time, or when the homogeneous state is
oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to
higher-order spatial systems are also included as a way of demonstrating the
generality of the approach
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