Roberto Ferretti

Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes

The convergence properties of a class of high-order semi-Lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions. We review the general convergence results for discrete-time approximation schemes belonging to that class and we prove some a priori estimates in

Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations

L^\infty

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

and L2 for the rate of convergence of fully discrete schemes. We prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions. Several examples of schemes and numerical tests are presented.

Convergence of Semi-Lagrangian Approximations to Convex Hamilton-Jacobi Equations under (Very) Large Courant Numbers

This largely self-contained book provides a unified framework of semi-Lagrangian strategy for the approximation of hyperbolic PDEs, with a special focus on Hamilton-Jacobi equations. The authors provide a rigorous discussion of the theory of viscosity solutions and the concepts underlying the construction and analysis of difference schemes; they then proceed to high-order semi-Lagrangian schemes and their applications to problems in fluid dynamics, front propagation, optimal control, and image processing. The developments covered in the text and the references come from a wide range of literature.

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

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.

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

We consider a class of semi-Lagrangian high-order approximation schemes for convex Hamilton--Jacobi equations. In this framework, we prove that under certain restrictions on the relationship between

Monotone Numerical Schemes and Feedback Construction for Hybrid Control Systems

\Delta x

Semi-Lagrangian Methods for Parabolic Problems in Divergence Form

and

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

\Delta t

Flux form Semi-Lagrangian methods for parabolic problems

, the sequence of approximate solutions is uniformly Lipschitz continuous and hence, by consistency, that it converges to the exact solution. The argument is suitable for most reconstructions of interest, including high-order polynomials and ENO reconstructions.