8 research outputs found
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)
A stroboscopic averaging algorithm for highly oscillatory delay problems
We propose and analyze a heterogenous multiscale method for the efficient
integration of constant-delay differential equations subject to fast periodic
forcing. The stroboscopic averaging method (SAM) suggested here may provide
approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a
computational effort that grows like \(H^{-1}\) (the inverse of the
stepsize), uniformly in the forcing frequency Omega
From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials
We present a multiscale integrator for Hamiltonian systems with slowly
varying quadratic stiff potentials that uses coarse timesteps (analogous to
what the impulse method uses for constant quadratic stiff potentials). This
method is based on the highly-non-trivial introduction of two efficient
symplectic schemes for exponentiations of matrices that only require O(n)
matrix multiplications operations at each coarse time step for a preset small
number n. The proposed integrator is shown to be (i) uniformly convergent on
positions; (ii) symplectic in both slow and fast variables; (iii) well adapted
to high dimensional systems. Our framework also provides a general method for
iteratively exponentiating a slowly varying sequence of (possibly high
dimensional) matrices in an efficient way
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
Numerical stroboscopic averaging for ODEs and DAEs
International audienceThe stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt = f(y; t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt = F(Y ) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LISP algorithms of Kirchgraber and multirevolution methods
The heterogeneous multiscale method
The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000209143400001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701MathematicsSCI(E)49ARTICLE1-872