852 research outputs found
Time-average on the numerical integration of nonautonomous differential equations
[EN] In this work we show how to numerically integrate nonautonomous differential equations by solving alternate time-averaged differential equations. Given a quadrature rule of order 2s or higher for s = 1, 2, . . . , we show how to build a differential equation with an averaged vector field that is a polynomial function of degree s - 1 in the independent variable, t, and whose solution after one time step agrees with the solution of the original differential equation up to order 2s. Then, any numerical scheme can be used to solve this alternate averaged equation where the vector field is always evaluated at the chosen quadrature rule. We show how to use the Magnus series expansion, adapted to nonlinear problems, to build the formal solution, and this result is valid for any choice of the quadrature rule. This formal solution can be used to build new schemes that must agree with it up to the desired order. For example, we show how to build commutator-free methods from previous results in the literature. All methods can also be written in terms of moment integrals, and each integral can be computed using different quadrature rules. This procedure allows us to build tailored methods for different classes of problems. We illustrate the time-averaged procedure and its efficiency in solving several problems.This work was funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Blanes Zamora, S. (2018). Time-average on the numerical integration of nonautonomous differential equations. SIAM Journal on Numerical Analysis. 56(4):2513-2536. https://doi.org/10.1137/17M1156150S2513253656
A Theoretical Framework for Lagrangian Descriptors
This paper provides a theoretical background for Lagrangian Descriptors
(LDs). The goal of achieving rigourous proofs that justify the ability of LDs
to detect invariant manifolds is simplified by introducing an alternative
definition for LDs. The definition is stated for -dimensional systems with
general time dependence, however we rigorously prove that this method reveals
the stable and unstable manifolds of hyperbolic points in four particular 2D
cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic
saddle point for nonlinear autonomous systems, a hyperbolic saddle point for
linear nonautonomous systems and a hyperbolic saddle point for nonlinear
nonautonomous systems. We also discuss further rigorous results which show the
ability of LDs to highlight additional invariants sets, such as -tori. These
results are just a simple extension of the ergodic partition theory which we
illustrate by applying this methodology to well-known examples, such as the
planar field of the harmonic oscillator and the 3D ABC flow. Finally, we
provide a thorough discussion on the requirement of the objectivity
(frame-invariance) property for tools designed to reveal phase space structures
and their implications for Lagrangian descriptors
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
On the Gap between Random Dynamical Systems and Continuous Skew Products
AMS 2000 subject classification: primary 37-02, 37B20, 37H05; secondary 34C27, 37A20.We review the recent notion of a nonautonomous dynamical system (NDS), which has been introduced as an abstraction of both random dynamical systems and continuous skew product flows. Our focus is on fundamental analogies and discrepancies brought about by these two classes
of NDS. We discuss base dynamics mainly through almost periodicity and almost automorphy, and we emphasize the importance of these concepts for NDS which are generated by differential and difference equations. Nonautonomous dynamics is presented by means of representative examples. We also mention several natural yet unresolved questions
Transit times and mean ages for nonautonomous and autonomous compartmental systems
We develop a theory for transit times and mean ages for nonautonomous
compartmental systems. Using the McKendrick-von F\"orster equation, we show
that the mean ages of mass in a compartmental system satisfy a linear
nonautonomous ordinary differential equation that is exponentially stable. We
then define a nonautonomous version of transit time as the mean age of mass
leaving the compartmental system at a particular time and show that our
nonautonomous theory generalises the autonomous case. We apply these results to
study a nine-dimensional nonautonomous compartmental system modeling the
terrestrial carbon cycle, which is a modification of the Carnegie-Ames-Stanford
approach (CASA) model, and we demonstrate that the nonautonomous versions of
transit time and mean age differ significantly from the autonomous quantities
when calculated for that model
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