29 research outputs found
A Posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrodinger equation
We derive optimal order a posteriori error estimates for fully discrete
approximations of linear Schr\"odinger-type equations, in the
norm. For the discretization in time we use the Crank-Nicolson
method, while for the space discretization we use finite element spaces that
are allowed to change in time. The derivation of the estimators is based on a
novel elliptic reconstruction that leads to estimates which reflect the
physical properties of Schr\"odinger equations. The final estimates are
obtained using energy techniques and residual-type estimators. Various
numerical experiments for the one-dimensional linear Schr\"odinger equation in
the semiclassical regime, verify and complement our theoretical results. The
numerical implementations are performed with both uniform partitions and
adaptivity in time and space. For adaptivity, we further develop and analyze an
existing time-space adaptive algorithm to the cases of Schr\"odinger equations.
The adaptive algorithm reduces the computational cost substantially and
provides efficient error control for the solution and the observables of the
problem, especially for small values of the Planck constant
On a selection principle for multivalued semiclassical flows
We study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture
Modulation instability and convergence of the random-phase approximation for stochastic sea states
The nonlinear Schrödinger equation is widely used as an approximate model for the evolution in time of the water wave envelope. In the context of simulating ocean waves, initial conditions are typically generated from a measured power spectrum using the random-phase approximation, and periodized on an interval of length L. It is known that most realistic ocean waves power spectra do not exhibit modulation instability, but the most severe ones do; it is thus a natural question to ask whether the periodized random-phase approximation has the correct stability properties. In this work, we specify a random-phase approximation scaling, so that, in the limit of L→∞ ,the stability properties of the periodized problem are identical to those of the continuous power spectrum on the infinite line. Moreover, it is seen through concrete examples that using a too short computational domain can completely suppress the modulation instability.Peer reviewe
Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations
We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank–Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time
Regularized semiclassical limits:linear flows with infinite Lyapunov exponents
Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth
potentials give rise to ill-posed formal semiclassical limits. These problems
have attracted a lot of attention in the last few years, as a proxy for the
treatment of eigenvalue crossings, i.e. general systems. It has recently been
shown that the semiclassical limit for conical singularities is in fact
well-posed, as long as the Wigner measure (WM) stays away from singular saddle
points. In this work we develop a family of refined semiclassical estimates,
and use them to derive regularized transport equations for saddle points with
infinite Lyapunov exponents, extending the aforementioned recent results. In
the process we answer a related question posed by P. L. Lions and T. Paul in
1993. If we consider more singular potentials, our rigorous estimates break
down. To investigate whether conical saddle points, such as , admit a
regularized transport asymptotic approximation, we employ a numerical solver
based on posteriori error control. Thus rigorous upper bounds for the
asymptotic error in concrete problems are generated. In particular, specific
phenomena which render invalid any regularized transport for are
identified and quantified. In that sense our rigorous results are sharp.
Finally, we use our findings to formulate a precise conjecture for the
condition under which conical saddle points admit a regularized transport
solution for the WM
Impaired Insulin Profiles Following a Single Night of Sleep Restriction: The Impact of Acute Sprint Interval Exercise
Experimental sleep restriction (SR) has demonstrated reduced insulin sensitivity in healthy individuals. Exercise is well-known to be beneficial for metabolic health. A single bout of exercise has the capacity to increase insulin sensitivity for up to 2 days. Therefore, the current study aimed to determine if sprint interval exercise could attenuate the impairment in insulin sensitivity after one night of SR in healthy males. Nineteen males were recruited for this randomized crossover study which consisted of four conditions—control, SR, control plus exercise, and sleep restriction plus exercise. Time in bed was 8 hr (2300–0700) in the control conditions and 4 hr (0300–0700) in the SR conditions. Conditions were separated by a 1-week entraining period. Participants slept at home, and compliance was assessed using wrist actigraphy. Following the night of experimental sleep, participants either conducted sprint interval exercise or rested for the equivalent duration. An oral glucose tolerance test was then conducted. Blood samples were obtained at regular intervals for measurement of glucose and insulin. Insulin concentrations were higher in SR than control (p = .022). Late-phase insulin area under the curve was significantly lower in sleep restriction plus exercise than SR (862 ± 589 and 1,267 ± 558; p = .004). Glucose area under the curve was not different between conditions (p = .207). These findings suggest that exercise improves the late postprandial response following a single night of SR