Elisabetta Carlini

A Weighted Essentially Nonoscillatory, Large Time-Step Scheme for Hamilton--Jacobi Equations

We investigate the application of weighted essentially nonoscillatory (WENO) reconstructions to a class of semi-Lagrangian schemes for first order time-dependent Hamilton--Jacobi equations. In particular, we derive a general form of the scheme, study sufficient conditions for its convergence with high-order reconstructions, and perform numerical tests to study its efficiency. In addition, we prove that the weights of the WENO interpolants are positive for any order.

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

We present a new fast-marching algorithm for an eikonal equation with a velocity-changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the fast-marching method in two respects. The first is that the new scheme can deal with a time-dependent velocity, and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm, and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in

Convergence of a large time-step scheme for mean curvature motion

\mathbb{R}^2

A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

.

A Generalized Fast Marching Method for Dislocation Dynamics

We propose a new scheme for the level set approximation of motion by mean curvature (MCM). The scheme originates from a representation formula recently given by Soner and Touzi, which allows us to construct large time-step, Godunov-type schemes. One such scheme is presented and its consistency is analyzed. We also provide and discuss some numerical tests.

A Time—Adaptive Semi—Lagrangian Approximation to Mean Curvature Motion

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.

A Semi-Lagrangian Scheme for a Modified Version of the Hughes’ Model for Pedestrian Flow

In this paper, we consider a generalized fast marching method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hypersurface in

A Non-Monotone Fast Marching Scheme for a Hamilton-Jacobi Equation Modelling Dislocation Dynamics

\mathbb{R}^N

A Generalized Fast Marching Method on Unstructured Triangular Meshes

(with

A semi-Lagrangian scheme with radial basis approximation for surface reconstruction

N=2