# Luigi BrugnanoUniversity of Florence | UNIFI · Dipartimento di Matematica e Informatica "Ulisse Dini"

Luigi Brugnano

Computing Sciences, University of Bari, Italy

## About

185

Publications

24,005

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3,510

Citations

Citations since 2016

Introduction

In the last years, my research interests focused on the so called Geometric Integration. Namely, the study of numerical methods able to reproduce relevant geometric properties of the continuous dynamical system, induced by a given class of differential problems, into the discrete one induced by the numerical method used for their solution.

Additional affiliations

November 2001 - present

January 1996 - October 2001

November 1992 - December 1995

Education

November 1981 - November 1985

**University of Bari, Italy**

Field of study

- Computing Sciences

## Publications

Publications (185)

In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differe...

In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than the one based on the expansion of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving corresponding fractional differential equations. A few numerical illustrations are reported.

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) . Their derivation relies on the expansion of the vector field along a suitable orthonormal basis. Interestingly, this approach can be extended to cope wi...

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope w...

In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than that using the expression of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving corresponding fractional differential equations. A few numerical examples are reported.

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In this review paper, we recall this aspect and e...

The COVID-19 epidemic hit Italy, starting from the northern regions, in late February 2020. Its diffusion along the peninsula was quite inhomogeneous, at the moment of lockdown, imposed starting from 11 March 2020. Consequently, standard models, like SIR, fail to provide accurate forecast for this epidemic. We here recall the main facts about a mul...

Recently, spectral methods in time for the numerical solution of many kinds of differential equations have been considered and analysed. Here, the use of spectral methods in time is extended to cope with Delay Differential Equations (DDEs), where they also proved to be very effective. In particular, we show an application to the mrSI2R2 model used...

In this paper we are concerned with numerical methods for the one-sided event location in discontinuous differential problems, whose event function is nonlinear (in particular, of polynomial type). The original problem is transformed into an equivalent Poisson problem, which is effectively solved by suitably adapting a recently devised class of ene...

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge-Kutta methods, which is here recalled and revisited for general...

In this paper we are concerned with numerical methods for the one-sided event location in discontinuous differential problems, whose event function is nonlinear (in particular, of polynomial type). The original problem is transformed into an equivalent Poisson problem, which is effectively solved by suitably adapting a recently devised class of ene...

In this paper, we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some...

In this paper, we define arbitrarily high-order energy-conserving methods for Hamiltonian systems with quadratic holonomic constraints. The derivation of the methods is made within the so-called line integral framework. Numerical tests to illustrate the theoretical findings are presented.

These are the lecture notes (in Italian) of a course held in Perugia, Italy, during the summer 2002. They concern the basic facts on the iterative solution of linear systems. The course is self-contained and requires only basic knowledge of numerical linear algebra. This is a corrected version of the original lecture notes, and are dedicated to the...

We implement the PaperRank and AuthorRank indices introduced in [Amodio & Brugnano, 2014] in the Scopus database, in order to highlight quantitative and qualitative information that the bare number of citations and/or the h-index of an author are unable to provide. In addition to this, the new indices can be cheaply updated in Scopus, since this ha...

In this paper we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some...

In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is further extended to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be...

The paper concerns a new forecast model that includes the class of undiagnosed infected people, and has a multiregion extension, to cope with the in‐time and in‐space heterogeneity of an epidemic. The model is applied to the SARS‐CoV2 (COVID‐19) pandemic that, starting from the end of February 2020, began spreading along the Italian peninsula, by f...

We implement the PaperRank and AuthorRank indices introduced in [Amodio & Brugnano, 2014] in the Scopus database, in order to highlight quantitative and qualitative information that the bare number of citations and/or the h-index of an author are unable to provide. In addition to this, the new indices can be cheaply updated in Scopus, since this ha...

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite pow...

Recently, Hamiltonian Boundary Value Methods (HBVMs), have been used as spectral methods in time for effectively solving multi-frequency, highly-oscillatory and/or stiffly-oscillatory problems. A complete analysis of their use in such a fashion has been also carried out, providing a theoretical framework explaining their effectiveness. We report he...

Background. The paper concerns the SARS-CoV2 (COVID-19) pandemic that, starting from the end of February 2020, began spreading along the Italian peninsula, by first attacking small communities in north regions, and then extending to the center and south of Italy, including the two main islands. Objective. The creation of a forecast model that manag...

This paper focuses on highly efficient numerical methods for solving space-fractional diffusion equations. By combining the fourth-order quasi-compact difference scheme and boundary value methods, a class of quasi-compact boundary value methods are constructed. In order to accelerate the convergence rate of this class of methods, the Kronecker prod...

Gyrocenter dynamics of charged particles plays a fundamental role in plasma physics. In particular, accuracy and conservation of energy are important features for correctly performing long-time simulations. For this purpose, we here propose arbitrarily high-order energy conserving methods for its simulation. The analysis and the efficient implement...

Recently, the numerical solution of stiffly/highly oscillatory Hamiltonian problems has been attacked by using Hamiltonian boundary value methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective eve...

In this paper, we study the numerical solution of Manakov systems by using a spectrally accurate Fourier decomposition in space, coupled with a spectrally accurate time integration. This latter relies on the use of spectral Hamiltonian Boundary Value Methods. The used approach allows to conserve all the physical invariants of the systems. Some nume...

In this note, we describe a simple generalization of the basic SIR model for epidemic, in case of a multi-region scenario, to be used for predicting the COVID-19 epidemic spread in Italy.

This paper deals with the numerical computation and analysis for a class of two-dimensional time–space fractional convection–diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the converg...

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities symbolized by the positive and negative finite powers of the rad...

We introduce a dynamic precision floating-point arithmetic based on the Infinity Computer. This latter is a computational platform which can handle both infinite and infinitesimal quantities epitomized by the positive and negative finite powers of the symbol Open image in new window, which acts as a radix in a corresponding positional numeral syste...

Gyrocenter dynamics of charged particles plays a fundamental role in plasma physics. In particular, accuracy and conservation of energy are important features for correctly performing long-time simulations. For this purpose, we here propose arbitrarily high-order energy conserving methods for its simulation. The analysis and the efficient implement...

In this paper, we study the numerical solution of Manakov systems by using a spectrally accurate Fourier decomposition in space, coupled with a spectrally accurate time integration. This latter relies on the use of spectral Hamiltonian boundary Value Methods. The used approach allows to conserve all the physical invariants of the systems. Some nume...

In recent years, the class of energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) has been devised for numerically solving Hamiltonian problems. In this short note, we study their natural formulation as continuous-stage Runge-Kutta(-Nyström) methods, which allows a deeper insight in the methods.

Recently, the numerical solution of multi-frequency, highly oscillatory Hamiltonian problems has been attacked by using Hamiltonian boundary value methods (HBVMs) as spectral methods in time. When the problem derives from the space semi-discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stif...

Recently, Hamiltonian Boundary Value Methods (HBVMs), have been used for effectively solving multi-frequency, highly-oscillatory and/or stiffly-oscillatory problems. We here report a few examples showing that, when numerically solving Hamiltonian PDEs, such methods, if coupled with a spectrally accurate space semi-discretization, are able to provid...

In this note, we extend a technique recently used to devise a novel class of geometric integrators named Hamil-tonian Boundary Value Methods, to cope with nonlinear fractional differential equations. The approach relies on a truncated Fourier expansion of the vector field which yields a modified problem that can be suitably handled on a computer. A...

you can download the article, until August 30 2019, here:
https://authors.elsevier.com/a/1ZNKX508HiGWu

In recent years, the class of energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) has been devised for numerically solving Hamiltonian problems. In this short note, we study their natural formulation as continuous-stage Runge-Kutta methods, which allows a deeper insight in the methods.

Multi-frequency, highly oscillatory Hamiltonian problems derive from the mathematical modelling of many real-life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems. We present algorithms to select the parameters of the methods that al...

In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries eq...

In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schr\"odinger equation, and the Korteweg-de Vr...

In this paper we study arbitrarily high-order energy-conserving methods for simulating the dynamics of a charged particle. They are derived and studied within the framework of Line Integral Methods (LIMs), previously used for defining Hamiltonian Boundary Value Methods (HBVMs), a class of energy-conserving Runge-Kutta methods for Hamiltonian proble...

In this paper we study the geometric numerical solution of the so called “good” Boussinesq equation. This goal is achieved by using a convenient space semi‐discretization, able to preserve the corresponding Hamiltonian structure, then using energy‐conserving Runge–Kutta methods in the Hamiltonian boundary value method class for the time integration...

The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at...

Recently, the numerical solution of stiffly/highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective eve...

In this paper we study the efficient solution of the well-known Korteweg–de Vries equation, equipped with periodic boundary conditions. A Fourier–Galerkin space semi-discretization at first provides a large-size Hamiltonian ODE problem, whose solution in time is then carried out by means of energy-conserving methods in the HBVM class (Hamiltonian B...

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be sti...

In this paper we study the geometric solution of the so called "good" Boussinesq equation. This goal is achieved by using a convenient space semi-discretization, able to preserve the corresponding Hamiltonian structure, then using energy-conserving Runge-Kutta methods in the HBVM class for the time integration. Numerical tests are reported, confirm...

In this paper we are concerned with the predictor-corrector implementation of a class of numerical integrators which can be considered as energy-conserving variants of the Gauss collocation methods.

In this paper, we study the efficient solution of the nonlinear Schrödinger equation with wave operator, subject to periodic boundary conditions. In such a case, it is known that its solution conserves a related functional. By using a Fourier expansion in space, the problem is at first casted into Hamiltonian form, with the same Hamiltonian functio...

In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used....

The numerical solution of highly oscillatory initial value problems of second order with a unique high frequency is considered. New methods based on Fourier approximations are proposed. These methods can integrate the problems with reasonable stepsizes not dependent on the size of the frequency.

In this paper we study the use of energy-conserving methods, in the class of Hamiltonian Boundary Value Methods, for the numerical solution of the nonlinear Schrödinger equation.

In this paper we are concerned with the analysis of a class of geometric integrators, at first devised in [14, 18], which can be regarded as an energy-conserving variant of Gauss collocation methods. With these latter they share the property of conserving quadratic first integrals but, in addition, they also conserve the Hamiltonian function itself...

In this paper, we further develop recent results in the numerical solution of Hamiltonian partial differential equations (PDEs) (Brugnano et al., 2015), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We shall use HBVMs for solving...

In this paper we extend the application of Hamiltonian Boundary Value Methods (HBVMs), a class of energy-conserving Runge-Kutta methods for Hamiltonian problems, to the numerical solution of Hamiltonian systems with holonomic constraints. The extension is obtained through a straightforward use of the so called line integral approach on which the me...

In this paper, we study recent results in the numerical solution of Hamiltonian partial differential equations (PDEs), by means of energy-conserving methods in the class of Line Integral Methods , in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We show that the use of energy-conserving methods, able to conse...

The family of EQUIP (Energy and QUadratic Invariants Preserving) methods for Hamiltonian systems is here recasted in the framework of Line Integral Methods, in order to provide a more efficient discrete problem.

We sketch out the use of the line integral as a tool to devise numerical methods suitable for conservative and, in particular, Hamiltonian problems. The monograph [3] presents the fundamental theory on line integral methods and this short note aims at exploring some aspects and results emerging from their study.

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

This book deals with the numerical solution of differential problems within
the framework of Geometric Integration, a branch of numerical analysis which
aims to devise numerical methods able to reproduce, in the discrete solution,
relevant geometric properties of the continuous vector field. Among them, a
paramount role is played by the so called c...

In this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the prescribed boundary conditions. We relate the conservation properties of the original problem to those of its semi-d...

The numerical solution of Hamiltonian PDEs has been the subject of many investigations in the last years, specially concerning the use of multi-symplectic methods. We shall here be concerned with the use of energy-conserving methods in the HBVMs class, when a spectral space discretization is considered.

In this paper we show that energy conserving methods, in particular those in the class of Hamiltonian Boundary Value Methods, can be conveniently used for the numerical solution of Hamiltonian Partial Differential Equations, after a suitable space semi-discretization.

In this paper we define a class of modified line integral methods, which are a suitable modification of energy conserving methods in the HBVMs class, able to cope with conservative problems possessing multiple invariants. The analysis of the methods is sketched, along with some numerical tests.

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

In this paper we discuss energy conservation issues related to the numerical
solution of the nonlinear wave equation. As is well known, this problem can be
cast as a Hamiltonian system that may be autonomous or not, depending on the
specific boundary conditions at hand. We relate the conservation properties of
the original problem to those of its s...

In this paper we discuss energy conservation issues related to the numerical
solution of the nonlinear wave equation, when a Fourier expansion is considered
for the space discretization. The obtained semi-discrete problem is then solved
in time by means of energy-conserving Runge-Kutta methods in the HBVMs class.

This special issue of the Communications in Nonlinear Science and Numerical Simulation contains a collection of research papers dealing with various problems of nonlinearity in physics, pure and applied mathematics, and computer science. It also considers numerical techniques and problems arising when natural phenomena are modelled on computers. Th...

We consider the issue of energy conservation in the numerical solution
of Hamiltonian systems coupled with boundary conditions and discuss a
few examples arising from astrodynamics.

We introduce new methods for the numerical solution of general Hamiltonian
boundary value problems. The main feature of the new formulae is to produce
numerical solutions along which the energy is precisely conserved, as is the
case with the analytical solution. We apply the methods to locate periodic
orbits in the circular restricted three body pr...