498 research outputs found
On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations
We present a convergence proof for higher order implementations of the
projective integration method (PI) for a class of deterministic multi-scale
systems in which fast variables quickly settle on a slow manifold. The error is
shown to contain contributions associated with the length of the microsolver,
the numerical accuracy of the macrosolver and the distance from the slow
manifold caused by the combined effect of micro- and macrosolvers,
respectively. We also provide stability conditions for the PI methods under
which the fast variables will not diverge from the slow manifold. We
corroborate our results by numerical simulations.Comment: 43 pages, 7 figures; accepted for publication in the Journal of
Computational and Applied Mathematic
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
Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit
We investigate a projective integration scheme for a kinetic equation in the
limit of vanishing mean free path, in which the kinetic description approaches
a diffusion phenomenon. The scheme first takes a few small steps with a simple,
explicit method, such as a spatial centered flux/forward Euler time
integration, and subsequently projects the results forward in time over a large
time step on the diffusion time scale. We show that, with an appropriate choice
of the inner step size, the time-step restriction on the outer time step is
similar to the stability condition for the diffusion equation, whereas the
required number of inner steps does not depend on the mean free path. We also
provide a consistency result. The presented method is asymptotic-preserving, in
the sense that the method converges to a standard finite volume scheme for the
diffusion equation in the limit of vanishing mean free path. The analysis is
illustrated with numerical results, and we present an application to the
Su-Olson test
Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position
Impulse methods are generalized to a family of integrators for Langevin
systems with quadratic stiff potentials and arbitrary soft potentials. Uniform
error bounds (independent from stiff parameters) are obtained on integrated
positions allowing for coarse integration steps. The resulting integrators are
explicit and structure preserving (quasi-symplectic for Langevin systems)
Projective Integration Methods in the Runge-Kutta Framework and the Extension to Adaptivity in Time
Projective Integration methods are explicit time integration schemes for
stiff ODEs with large spectral gaps. In this paper, we show that all existing
Projective Integration methods can be written as Runge-Kutta methods with an
extended Butcher tableau including many stages. We prove consistency and order
conditions of the Projective Integration methods using the Runge-Kutta
framework. Spatially adaptive Projective Integration methods are included via
partitioned Runge-Kutta methods. New time adaptive Projective Integration
schemes are derived via embedded Runge-Kutta methods and step size variation
while their accuracy, stability, convergence, and error estimators are
investigated analytically and numerically
Optimal stability polynomials for numerical integration of initial value problems
We consider the problem of finding optimally stable polynomial approximations
to the exponential for application to one-step integration of initial value
ordinary and partial differential equations. The objective is to find the
largest stable step size and corresponding method for a given problem when the
spectrum of the initial value problem is known. The problem is expressed in
terms of a general least deviation feasibility problem. Its solution is
obtained by a new fast, accurate, and robust algorithm based on convex
optimization techniques. Global convergence of the algorithm is proven in the
case that the order of approximation is one and in the case that the spectrum
encloses a starlike region. Examples demonstrate the effectiveness of the
proposed algorithm even when these conditions are not satisfied
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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