13,609 research outputs found
On Monotonicity and Propagation of Order Properties
In this paper, a link between monotonicity of deterministic dynamical systems
and propagation of order by Markov processes is established. The order
propagation has received considerable attention in the literature, however,
this notion is still not fully understood. The main contribution of this paper
is a study of the order propagation in the deterministic setting, which
potentially can provide new techniques for analysis in the stochastic one. We
take a close look at the propagation of the so-called increasing and increasing
convex orders. Infinitesimal characterisations of these orders are derived,
which resemble the well-known Kamke conditions for monotonicity. It is shown
that increasing order is equivalent to the standard monotonicity, while the
class of systems propagating the increasing convex order is equivalent to the
class of monotone systems with convex vector fields. The paper is concluded by
deriving a novel result on order propagating diffusion processes and an
application of this result to biological processes.Comment: Part of the paper is to appear in American Control Conference 201
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft
Non-monotonicity of the frictional bimaterial effect
Sliding along frictional interfaces separating dissimilar elastic materials
is qualitatively different from sliding along interfaces separating identical
materials due to the existence of an elastodynamic coupling between interfacial
slip and normal stress perturbations in the former case. This bimaterial
coupling has important implications for the dynamics of frictional interfaces,
including their stability and rupture propagation along them. We show that
while this bimaterial coupling is a monotonically increasing function of the
bimaterial contrast, when it is coupled to interfacial shear stress
perturbations through a friction law, various physical quantities exhibit a
non-monotonic dependence on the bimaterial contrast. In particular, we show
that for a regularized Coulomb friction, the maximal growth rate of unstable
interfacial perturbations of homogeneous sliding is a non-monotonic function of
the bimaterial contrast, and provide analytic insight into the origin of this
non-monotonicity. We further show that for velocity-strengthening
rate-and-state friction, the maximal growth rate of unstable interfacial
perturbations of homogeneous sliding is also a non-monotonic function of the
bimaterial contrast. Results from simulations of dynamic rupture along a
bimaterial interface with slip-weakening friction provide evidence that the
theoretically predicted non-monotonicity persists in non-steady, transient
frictional dynamics.Comment: 14 pages, 5 figure
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