1,107 research outputs found
Equation-free modeling of evolving diseases: Coarse-grained computations with individual-based models
We demonstrate how direct simulation of stochastic, individual-based models
can be combined with continuum numerical analysis techniques to study the
dynamics of evolving diseases. % Sidestepping the necessity of obtaining
explicit population-level models, the approach analyzes the (unavailable in
closed form) `coarse' macroscopic equations, estimating the necessary
quantities through appropriately initialized, short `bursts' of
individual-based dynamic simulation. % We illustrate this approach by analyzing
a stochastic and discrete model for the evolution of disease agents caused by
point mutations within individual hosts. % Building up from classical SIR and
SIRS models, our example uses a one-dimensional lattice for variant space, and
assumes a finite number of individuals. % Macroscopic computational tasks
enabled through this approach include stationary state computation, coarse
projective integration, parametric continuation and stability analysis.Comment: 16 pages, 8 figure
Coarse-graining the dynamics of coupled oscillators
We present an equation-free computational approach to the study of the
coarse-grained dynamics of {\it finite} assemblies of {\it non-identical}
coupled oscillators at and near full synchronization. We use coarse-grained
observables which account for the (rapidly developing) correlations between
phase angles and oscillator natural frequencies. Exploiting short bursts of
appropriately initialized detailed simulations, we circumvent the derivation of
closures for the long-term dynamics of the assembly statistics.Comment: accepted for publication in Phys. Rev. Let
Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models
We introduce a general formulation for an implicit equation-free method in
the setting of slow-fast systems. First, we give a rigorous convergence result
for equation-free analysis showing that the implicitly defined coarse-level
time stepper converges to the true dynamics on the slow manifold within an
error that is exponentially small with respect to the small parameter measuring
time scale separation. Second, we apply this result to the idealized traffic
modeling problem of phantom jams generated by cars with uniform behavior on a
circular road. The traffic jams are waves that travel slowly against the
direction of traffic. Equation-free analysis enables us to investigate the
behavior of the microscopic traffic model on a macroscopic level. The standard
deviation of cars' headways is chosen as the macroscopic measure of the
underlying dynamics such that traveling wave solutions correspond to equilibria
on the macroscopic level in the equation-free setup. The collapse of the
traffic jam to the free flow then corresponds to a saddle-node bifurcation of
this macroscopic equilibrium. We continue this bifurcation in two parameters
using equation-free analysis.Comment: 35 page
Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized Relaxation Redistribution Method
In this paper, we introduce a fictitious dynamics for describing the only
fast relaxation of a stiff ordinary differential equation (ODE) system towards
a stable low-dimensional invariant manifold in the phase-space (slow invariant
manifold - SIM). As a result, the demanding problem of constructing SIM of any
dimensions is recast into the remarkably simpler task of solving a properly
devised ODE system by stiff numerical schemes available in the literature. In
the same spirit, a set of equations is elaborated for local construction of the
fast subspace, and possible initialization procedures for the above equations
are discussed. The implementation to a detailed mechanism for combustion of
hydrogen and air has been carried out, while a model with the exact
Chapman-Enskog solution of the invariance equation is utilized as a benchmark.Comment: accepted in J. Comp. Phy
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