16,174 research outputs found
An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry.\ud
We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities
Kinematic reduction of reaction-diffusion fronts with multiplicative noise: Derivation of stochastic sharp-interface equations
We study the dynamics of generic reaction-diffusion fronts, including pulses
and chemical waves, in the presence of multiplicative noise. We discuss the
connection between the reaction-diffusion Langevin-like field equations and the
kinematic (eikonal) description in terms of a stochastic moving-boundary or
sharp-interface approximation. We find that the effective noise is additive and
we relate its strength to the noise parameters in the original field equations,
to first order in noise strength, but including a partial resummation to all
orders which captures the singular dependence on the microscopic cutoff
associated to the spatial correlation of the noise. This dependence is
essential for a quantitative and qualitative understanding of fluctuating
fronts, affecting both scaling properties and nonuniversal quantities. Our
results predict phenomena such as the shift of the transition point between the
pushed and pulled regimes of front propagation, in terms of the noise
parameters, and the corresponding transition to a non-KPZ universality class.
We assess the quantitative validity of the results in several examples
including equilibrium fluctuations, kinetic roughening, and the noise-induced
pushed-pulled transition, which is predicted and observed for the first time.
The analytical predictions are successfully tested against rigorous results and
show excellent agreement with numerical simulations of reaction-diffusion field
equations with multiplicative noise.Comment: 17 pages, 6 figure
A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions
We present a numerical approximation method for linear diffusion-reaction
problems with possibly discontinuous Dirichlet boundary conditions. The
solution of such problems can be represented as a linear combination of
explicitly known singular functions as well as of an -regular part. The
latter part is expressed in terms of an elliptic problem with regularized
Dirichlet boundary conditions, and can be approximated by means of a Nitsche
finite element approach. The discrete solution of the original problem is then
defined by adding the singular part of the exact solution to the Nitsche
approximation. In this way, the discrete solution can be shown to converge of
second order with respect to the mesh size
Two-scale large deviations for chemical reaction kinetics through second quantization path integral
Motivated by the study of rare events for a typical genetic switching model
in systems biology, in this paper we aim to establish the general two-scale
large deviations for chemical reaction systems. We build a formal approach to
explicitly obtain the large deviation rate functionals for the considered
two-scale processes based upon the second-quantization path integral technique.
We get three important types of large deviation results when the underlying two
times scales are in three different regimes. This is realized by singular
perturbation analysis to the rate functionals obtained by path integral. We
find that the three regimes possess the same deterministic mean-field limit but
completely different chemical Langevin approximations. The obtained results are
natural extensions of the classical large volume limit for chemical reactions.
We also discuss its implication on the single-molecule Michaelis-Menten
kinetics. Our framework and results can be applied to understand general
multi-scale systems including diffusion processes
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