Luciano Lopez

Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach

In this work, we discuss some theoretical and numerical aspects of solving differential equations with discontinuous right-hand sides of Filippov type. In particular, (i) we propose second order corrections to the theory of Filippov, (ii) we provide a systematic and nonambiguous way to define the vector field on the intersection of several surfaces of discontinuity, and (iii) we propose, and implement, a numerical method to approximate a trajectory of systems with discontinuous right-hand sides and illustrate its performance on a few examples.

A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side

This work is dedicated to the memory of Donato Trigiante who has been the first teacher of Numerical Analysis of the second author. The authors remember Donato as a generous teacher, always ready to discuss with his students, able to give them profound and interesting suggestions. Here, we present a survey of numerical methods for differential systems with discontinuous right hand side. In particular, we will review methods where the discontinuities are detected by using an event function (so-called event driven methods) and methods where the discontinuities are located by controlling the local errors (so-called time-stepping methods). Particular attention will be devoted to discontinuous systems of Filippovs type where sliding behavior on the discontinuity surface is allowed.

Boundary value methods and BV-stability in the solution of initial value problems

Abstract In this paper we consider boundary value techniques based on a three-term numerical method for solving initial value problems. The notions of BV-stability and BV-relative stability are introduced in order to clarify the conditions that a three-term scheme must satisfy for solving efficiently initial value problems. In particular we investigate the BV-stability of boundary value methods based on the mid-point rule, on the Simpson method, and on an Adams-type method. The problem of approximating the solution at the final point is approached and an error estimate at this point is given. Among the main features of the boundary value methods studied there is the possibility of employing the same method for an initial value problem with increasing and decreasing modes and the possibility of implementing efficiently boundary value methods on parallel computers.

Computation of the Exponential of Large Sparse Skew-Symmetric Matrices

In this paper we consider methods for evaluating both exp(A) and

The Use of the Factorization of Five-Diagonal Matrices by Tridiagonal Toeplitz Matrices

exp(\tau A)q_1

Runge-Kutta Type Methods Based on Geodesics for Systems of ODEs on the Stiefel Manifold

where

On the Low-Rank Approximation of Data on the Unit Sphere

{\rm exp}(\cdot)

On the continuous extension of Adams-Bashforth methods and the event location in discontinuous ODEs

is the exponential function, A is a sparse skew-symmetric matrix of large dimension, q1 is a given vector, and

Fundamental matrix solutions of piecewise smooth differential systems

\tau

Geometric Integration on Manifold of Square Oblique Rotation Matrices

is a scaling factor. The proposed method is based on two main steps: A is factorized into its tridiagonal form H by the well-known Lanczos iterative process, and then exp(A) is derived making use of an effective Schur decomposition of H. The procedure takes full advantage of the sparsity of A and of the decay behavior of exp(H). Several applications and numerical tests are also reported.