2,249 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)
Sparsing in Real Time Simulation
Modelling of mechatronical systems often leads to large DAEs with stiff components. In real time simulation neither implicit nor explicit methods can cope with such systems in an efficient way: explicit methods have to employ too small steps and implicit methods have to solve too large systems of equations. A solution of this general problem is to use a method that allows manipulations of the Jacobian by computing only those parts that are necessary for the stability of the method. Specifically, manipulation by sparsing aims at zeroing out certain elements of the Jacobian leading t a structure that can be exploited using sparse matrix techniques. The elements to be neglected are chosen by an a priori analysis phase that can be accomplished before the real-time simulaton starts. In this article a sparsing criterion for the linearly implicit Euler method is derived that is based on block diagnonalization and matrix perturbation theory
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
Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions
For trigonometric and modified trigonometric integrators applied to
oscillatory Hamiltonian differential equations with one or several constant
high frequencies, near-conservation of the total and oscillatory energies are
shown over time scales that cover arbitrary negative powers of the step size.
This requires non-resonance conditions between the step size and the
frequencies, but in contrast to previous results the results do not require any
non-resonance conditions among the frequencies. The proof uses modulated
Fourier expansions with appropriately modified frequencies.Comment: 26 page
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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