Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStructure-preserving numerical schemes for Lindblad equations
126
0
0.0
(
0
)
Added by
Yu Cao
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations, and carry out detailed error analysis and absolute stability analysis. Both error and absolute stability analysis are validated by numerical examples.
rate research
Read More
In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are formulated in a one-dimensional setting, here we consider exclusively the two-dimensional case. We prove that the proposed schemes preserve fundamental structural properties like nonnegativity of the solution without restriction on the size of the mesh and entropy dissipation. Moreover, all the methods presented here are at least second order accurate in the transient regimes and arbitrarily high order for large times in the hypothesis in which the flux vanishes at the stationary state. Suitable numerical tests will confirm the theoretical results.
We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space-time Lagrange multiplier to enforce the positivity with the Karush-Kuhn-Tucker (KKT) conditions. We then use a predictor-corrector approach to construct a class of positivity schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity, can be implemented with negligible cost. We also present a modification which allows us to construct schemes which, in addition to positivity preserving, is also mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization schemes. We establish stability results for our first- and second-order schemes under a general setting, and present ample numerical results to validate the new approach.
In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.
This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.
In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning rule. The equation contains a small parameter $varepsilon$ separating the time scales of learning and reacting behavior of the neural system, and an asymptotic limit model can be derived by letting $varepsilonto 0$, where the microscopic quasi-static states and the macroscopic evolution equation are coupled through the total firing rate. To handle the endowed flux-shift structure and the multi-scale dynamics in a unified framework, we propose a numerical scheme for this equation that is mass conservative, unconditionally positivity preserving, and asymptotic preserving. We provide extensive numerical tests to verify the schemes properties and carry out a set of numerical experiments to investigate the models learning ability, and explore the solutions behavior when the neural network is excitatory.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
