Abstract
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.
Highlights
The numerical solution of differential problems in the form ẏ(t) = f (y(t)), t ≥ 0, y (0 ) = y 0 ∈ D ⊆ Rm, (1)is needed in a variety of applications
The paper is organized as follows: in Section 2 we shall deal with the numerical solution of Hamiltonian problems; Poisson problems are considered in Section 3; constrained Hamiltonian problems are studied in Section 4; Hamiltonian partial differential equations (PDEs) are considered in Section 5; highly oscillatory problems are briefly discussed in Section 6; at last, Section 7 contains some concluding remarks
We have reviewed the main facts concerning the numerical solution of conservative problems within the framework of line integral methods
Summary
The path σ satisfying (5) and (8) defines a discrete line integral method providing an approximation y1 to y(h) such that C (y1 ) ≈ C (y0 ), within the accuracy of the quadrature rule. The main reference on line integral methods is given by the monograph [1] With these premises, the paper is organized as follows: in Section 2 we shall deal with the numerical solution of Hamiltonian problems; Poisson problems are considered in Section 3; constrained Hamiltonian problems are studied in Section 4; Hamiltonian partial differential equations (PDEs) are considered in Section 5; highly oscillatory problems are briefly discussed in Section 6; at last, Section 7 contains some concluding remarks
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