606 research outputs found
Simulating Stochastic Inertial Manifolds by a Backward-Forward Approach
A numerical approach for the approximation of inertial manifolds of
stochastic evolutionary equations with multiplicative noise is presented and
illustrated. After splitting the stochastic evolutionary equations into a
backward and a forward part, a numerical scheme is devised for solving this
backward-forward stochastic system, and an ensemble of graphs representing the
inertial manifold is consequently obtained. This numerical approach is tested
in two illustrative examples: one is for a system of stochastic ordinary
differential equations and the other is for a stochastic partial differential
equation
Coarse Projective kMC Integration: Forward/Reverse Initial and Boundary Value Problems
In "equation-free" multiscale computation a dynamic model is given at a fine,
microscopic level; yet we believe that its coarse-grained, macroscopic dynamics
can be described by closed equations involving only coarse variables. These
variables are typically various low-order moments of the distributions evolved
through the microscopic model. We consider the problem of integrating these
unavailable equations by acting directly on kinetic Monte Carlo microscopic
simulators, thus circumventing their derivation in closed form. In particular,
we use projective multi-step integration to solve the coarse initial value
problem forward in time as well as backward in time (under certain conditions).
Macroscopic trajectories are thus traced back to unstable, source-type, and
even sometimes saddle-like stationary points, even though the microscopic
simulator only evolves forward in time. We also demonstrate the use of such
projective integrators in a shooting boundary value problem formulation for the
computation of "coarse limit cycles" of the macroscopic behavior, and the
approximation of their stability through estimates of the leading "coarse
Floquet multipliers".Comment: Submitted to Journal of Computational Physic
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure
Global error analysis and inertial manifold reduction
Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to include rescaling of time. To determine the amplificatioh of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto-Sivashinsky equation. (C) 2016 Elsevier B.V. All rights reserved
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
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