Liviu I. Ignat

Numerical Dispersive Schemes for the Nonlinear Schrödinger Equation

We consider semidiscrete approximation schemes for the linear Schrodinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with

A Compactness Tool for the Analysis of Nonlocal Evolution Equations

L^2

FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates.

Inverse problem for the heat equation and the Schrödinger equation on a tree

In this paper we analyze the long time behavior of the solutions of a nonlocal diffusion-convection equation. We give a new compactness criterion in the Lebesgue spaces

A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

L^p((0,T)\times \Omega)

Asymptotic behavior of solutions to fractional diffusion–convection equations

and use it to obtain the first term in the asymptotic expansion of the solutions. Previous results of [J. Bourgain, H. Brezis, and P. Mironescu, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001] are used to obtain a compactness result in the spirit of the Aubin--Lions--Simon lemma.

A nonlocal convection–diffusion equation

We consider fully discrete schemes for the one-dimensional linear Schrodinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrodinger equation with nonlinearities which cannot be treated by energy methods.

Decay estimates for nonlocal problems via energy methods

In this paper, we establish global Carleman estimates for the heat and Schrodinger equations on a network. The heat equation is considered on a general tree and the Schrodinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. Schrodinger) equation from boundary measurements.

Dispersive properties of a viscous numerical scheme for the Schrödinger equation

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain

Refined asymptotic expansions for nonlocal diffusion equations

L^1