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 methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.
A general framework for solving differential equations Brugnano, Luigi; Iavernaro, Felice
Annali dell'Università di Ferrara. Scienze Matematiche/Annali dell'Università di Ferrara. Sezione 7: Scienze matematiche,
2022/11, Volume:
68, Issue:
2
Journal Article
Peer reviewed
Open access
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 with more general differential problems. In this paper we sketch this fact, by considering some relevant examples.
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 differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.
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 extend it to higher-order differential problems.
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 equation, to illustrate the main features of this novel approach.
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 hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.
The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community ...conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.
•A conservation property for the nonlinear Schrödinger equation with wave operator is studied.•A spectrally accurate space semi-discretization is considered.•A class of energy-conserving boundary ...value methods for the induced large-size Hamiltonian ODE problem is considered.•The efficient implementation of the methods is studied and numerical tests are reported.
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 functional. A Fourier–Galerkin space semi-discretization then provides a large-size Hamiltonian ODE problem, whose solution in time is carried out by means of energy-conserving methods in the HBVM class (Hamiltonian boundary value methods). The efficient implementation of the methods for the resulting problem is also considered and some numerical examples are reported.
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 implementation of the methods are fully described, and some numerical tests are reported.
Recently, the class of Hamiltonian Boundary Value Methods (HBVMs) has been introduced with the aim of preserving the energy associated with polynomial Hamiltonian systems (and, more in general, with ...all suitably regular Hamiltonian systems). However, many interesting problems admit other invariants besides the Hamiltonian function. It would be therefore useful to have methods able to preserve any number of independent invariants. This goal is achieved by generalizing the line-integral approach which HBVMs rely on, thus obtaining a number of generalizations which we collectively name Line Integral Methods. In fact, it turns out that this approach is quite general, so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial, such as a collocation method, as well as to any conservative problem. In particular, a completely conservative variant of both HBVMs and Gauss collocation methods is presented. Numerical experiments confirm the effectiveness of the proposed methods.
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