Luigi Brugnano

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

In this paper, we provide a simple framework to derive and analyse several classes of effective one-step methods. The framework consists in the discretization of a local Fourier expansion of the continuous problem. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge-Kutta methods is derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems. A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems, via their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.

Luigi Brugnano, Gianluca Frasca Caccia, Felice Iavernaro

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 semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schrödinger equation.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

We introduce a new family of symplectic integrators depending on a real parameter. When the paramer is zero, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null value of the parameter, the corresponding method remains symplectic and has order 2s-2: hence it may be interpreted as an order 2s-2 (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the free parameter may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting symplectic, energy conserving method shares the same order 2s as the generating Gauss formula.

Luigi Brugnano, Gianluca Frasca Caccia, Felice Iavernaro

In this paper we define an efficient implementation for the family of low-rank energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), recently defined in the last years. The proposed implementation relies on the particular structure of the Butcher matrix defining such methods, for which we can derive an efficient splitting procedure. The very same procedure turns out to be automatically suited for the efficient implementation of Gauss-Legendre collocation methods, since these methods are a special instance of HBVMs. The linear convergence analysis of the splitting procedure exhibits excellent properties, which are confirmed by a few numerical tests.

Luigi Brugnano, Felice Iavernaro, Juan I. Montijano et al.

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 stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

When numerically integrating canonical Hamiltonian systems, the long-term conservation of some of its invariants, among which the Hamiltonian function itself, assumes a central role. The classical approach to this problem has led to the definition of symplectic methods, among which we mention Gauss-Legendre collocation formulae. Indeed, in the continuous setting, energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations. However, this infinite process cannot be directly transferred to the discrete setting. By following a different approach, in this paper we describe a sequence of methods, sharing the same essential spectrum (and, then, the same essential properties), which are energy preserving starting from a certain element of the sequence on, i.e., after a finite number of steps.

Luigi Brugnano, Yajuan Sun

In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative problems, thus providing the class of Line Integral Methods (LIMs). In this paper we derive a further extension, which we name Enhanced Line Integral Methods (ELIMs), more tailored for Hamiltonian problems, allowing for the conservation of multiple invariants of the continuous dynamical system. The analysis of the methods is fully carried out and some numerical tests are reported, in order to confirm the theoretical achievements.

Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro

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 even when applied to a wider range of problems. Here we fill this gap by providing a thorough convergence analysis of the methods and confirm the theoretical results with the aid of a few numerical tests.

Luigi Brugnano, Gianmarco Gurioli, Chengjian Zhang

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, confirming the effectiveness of the proposed method.

Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro

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. We here reformulate the methods in a more convenient way, and propose a more refined analysis than that given in [18] also providing, as a by-product, a practical procedure for their implementation. A thorough comparison with the original Gauss methods is carried out by means of a few numerical tests solving Hamiltonian and Poisson problems.

Luigi Brugnano, Gianluca Frasca-Caccia, Felice Iavernaro

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 Schrödinger equation, and the Korteweg-de Vries equation, to illustrate the main features of this novel approach.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems. In the mainstream of this research, we have defined a new family of symplectic integrators depending on a real parameter. When it is zero, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null value of the parameter, the corresponding method remains symplectic and has order 2s-2: hence it may be interpreted as a symplectic perturbation of the Gauss method. Under suitable assumptions, it can be shown that the parameter a may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order 2s as the generating Gauss formula, and is able to preserve both energy and quadratic invariants.

Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro et al.

In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustrate theoretical findings.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. Recently, a new class of methods, named Hamiltonian Boundary Value Methods (HBVMs) has been introduced and analysed, which are able to exactly preserve polynomial Hamiltonians of arbitrarily high degree. We here study a further property of such methods, namely that of having, when cast as a Runge-Kutta method, a matrix of the Butcher tableau with the same spectrum (apart from the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. This, in turn, allows to elucidate the existing connections with classical Runge-Kutta collocation methods.

Pierluigi Amodio, Luigi Brugnano

Bibliometric indexes are customary used in evaluating the impact of scientific research, even though it is very well known that in different research areas they may range in very different intervals. Sometimes, this is evident even within a single given field of investigation making very difficult (and inaccurate) the assessment of scientific papers. On the other hand, the problem can be recast in the same framework which has allowed to efficiently cope with the ordering of web-pages, i.e., to formulate the PageRank of Google. For this reason, we call such problem the PaperRank problem, here solved by using a similar approach to that employed by PageRank. The obtained solution, which is mathematically grounded, will be used to compare the usual heuristics of the number of citations with a new one here proposed. Some numerical tests show that the new heuristics is much more reliable than the currently used ones, based on the bare number of citations. Moreover, we show that our model improves on recently proposed ones.

Luigi Brugnano, Felice Iavernaro

These are the lecture notes of a course given by the first author on December 27, 2012 - January 4, 2013, held at the Academy of Mathematics and Systems Science Chinese Academy of Sciences in Beijing.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

We report a few sumerical tests comparing some newly defined energy-preserving integrators and symplectic methods, using either constant and variable stepsize.

Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro et al.

The efficient numerical solution of fractional differential equations has been recently tackled through the definition of Fractional HBVMs (FHBVMs), a class of Runge-Kutta type methods. Corresponding Matlab (c) codes have been also made available on the internet, proving to be very competitive w.r.t. existing ones. However, so far, FHBVMs have been given for solving systems of fractional differential equations with the same order of fractional derivative, whereas the numerical solution of multi-order problems (i.e., problems in which different orders of fractional derivatives occur) has not been handled, yet. Due to their relevance in applications, in this paper we propose an extension of FHBVMs for addressing fractional multi-order problems, providing full details for such an approach. A corresponding Matlab (c) code, handling the case of two different fractional orders, is also made available, proving very effective for numerically solving these problems.

Luigi Brugnano, Gianluca Frasca Caccia, Felice Iavernaro

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.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction term is O(h^5) (h is the stepsize of integration). The key tools the new methods are based upon are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

Recently, a new family of integrators (Hamiltonian Boundary ValueMethods) has been introduced, which is able to precisely conserve the energy function of polynomial Hamiltonian systems and to provide a practical conservation of the energy in the non-polynomial case. We settle the definition and the theory of such methods in a more general framework. Our aim is on the one hand to give account of their good behavior when applied to general Hamiltonian systems and, on the other hand, to find out what are the optimal formulae, in relation to the choice of the polynomial basis and of the distribution of the nodes. Such analysis is based upon the notion of extended collocation conditions and the definition of discrete line integral, and is carried out by looking at the limit of such family of methods as the number of the so called silent stages tends to infinity.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

Hamiltonian Boundary Value Methods (in short, HBVMs) is a new class of numerical methods for the efficient numerical solution of canonical Hamiltonian systems. In particular, their main feature is that of exactly preserving, for the numerical solution, the value of the Hamiltonian function, when the latter is a polynomial of arbitrarily high degree. Clearly, this fact implies a practical conservation of any analytical Hamiltonian function. In this notes, we collect the introductory material on HBVMs contained in the HBVMs Homepage, available at http://web.math.unifi.it/users/brugnano/HBVM/index.html

Luigi Brugnano, Felice Iavernaro, Tiziana Susca

Hamiltonian Boundary Value Methods are a new class of energy preserving one step methods for the solution of polynomial Hamiltonian dynamical systems. They can be thought of as a generalization of collocation methods in that they may be defined by imposing a suitable set of extended collocation conditions. In particular, in the way they are described in this note, they are related to Gauss collocation methods with the difference that they are able to precisely conserve the Hamiltonian function in the case where this is a polynomial of any high degree in the momenta and in the generalized coordinates. A description of these new formulas is followed by a few test problems showing how, in many relevant situations, the precise conservation of the Hamiltonian is crucial to simulate on a computer the correct behavior of the theoretical solutions.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. Recently, a new class of methods, named "Hamiltonian Boundary Value Methods (HBVMs)" has been introduced and analysed, which are able to exactly preserve polynomial Hamiltonians of arbitrarily high degree. We here study a further property of such methods, namely that of having, when cast as Runge-Kutta methods, a matrix of the Butcher tableau with the same spectrum (apart the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. Moreover, this allows their efficient "blended" implementation, for solving the generated discrete problems.

Luigi Brugnano, Cecilia Magherini

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, order barriers for stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been made in order to derive GLMs with particular features, to be exploited for their efficient implementation. In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE-IVPs. In particular, this approach has been recently developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose theory is here completed and fully worked-out. Moreover, for each one of such methods, it is possible to define a corresponding Blended GLM which is equivalent to it from the point of view of the stability and order properties. These blended methods, in turn, allow the definition of efficient nonlinear splittings for solving the generated discrete problems. A few numerical tests, confirming the excellent potential of such blended methods, are also reported.

Pierluigi Amodio, Luigi Brugnano

In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.

Luigi Brugnano, Alessandra Sestini

We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the numerical solution of free-surface problems. In particular, we here study their application to the numerical solution of both the (linear) parabolic obstacle problem and the obstacle problem. We propose a class of effective semi-iterative Newton-type methods to find the exact solution of such piecewise linear systems. We prove that the semiiterative Newton-type methods have a global monotonic convergence property, i.e., the iterates converge monotonically to the exact solution in a finite number of steps. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Luigi Brugnano, Felice Iavernaro, Donato Trigiante

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. For example, it is well known that classical symplectic methods can only exactly preserve, at most, quadratic Hamiltonians. In this paper, a new family of methods, called "Hamiltonian Boundary Value Methods (HBVMs)", is introduced and analyzed. HBVMs are able to exactly preserve, in the discrete solution, Hamiltonian functions of polynomial type of arbitrarily high degree. These methods turn out to be symmetric, precisely A-stable, and can have arbitrarily high order. A few numerical tests confirm the theoretical results.

Luigi Brugnano, Francesca Mazzia, Donato Trigiante

The notion of stiffness, which originated in several applications of a different nature, has dominated the activities related to the numerical treatment of differential problems for the last fifty years. Contrary to what usually happens in Mathematics, its definition has been, for a long time, not formally precise (actually, there are too many of them). Again, the needs of applications, especially those arising in the construction of robust and general purpose codes, require nowadays a formally precise definition. In this paper, we review the evolution of such a notion and we also provide a precise definition which encompasses all the previous ones.