Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGradient Flow Approach to the Calculation of Stationary States on Nonlinear Quantum Graphs
126
0
0.0
(
0
)
Added by
Stefan Le Coz
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Christophe Besse
Ask ChatGPT about the research
No Arabic abstract
We introduce and implement a method to compute stationary states of nonlinear Schrodinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schrodinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.
rate research
Read More
Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schr{o}dinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schr{o}dinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schr{o}dinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.
We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connexions between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a selection of numerical simulations for a selection of connected graphs.
We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the traffic flow over the on-ramp and off-ramps. We approximate the problem using an upwind-type numerical scheme and we provide L^infty and BV estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.
A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics, and also as the gradient flow of a second-order information functional with respect to the $L^2$-Wasserstein metric. First, we prove global existence of weak solutions for initial conditions of finite entropy by means of the time-discrete minimizing movement scheme. Second, we calculate the linearization of the dynamics around the unique stationary solution, for which we can explicitly compute the entire spectrum. A key element in our approach is a particular relation between the entropy, the Fisher information and the second order functional that generates the gradient flow under consideration.
We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with the entropy analysis. The details brought forward by the nonlinear energy analysis allow us to pinpoint where the difference between the linear and nonlinear analysis originate. We find that the result from the linear analysis does not necessarily hold in the nonlinear case. The nonlinear analysis leads in general to a different minimal number of boundary conditions compared with the linear analysis. In particular, and contrary to the linear case, the magnitude of the flow does not influence the number of required boundary conditions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
