Energy-conserving methods for Hamiltonian boundary value problems and applications in astrodynamics

Abstract

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 problem by using their energy value rather than their period as input data. We also use the methods for solving optimal transfer problems in astrodynamics

High‐order energy‐preserving schemes for the improved Boussinesq equation

“…As is well known, the value of Hamiltonian function is a constant along the solution of (3.1). According to , this can be easily verified by line integral associated with the vector field evaluated along the path defined by the solution of (3.1), which equals the variation of along the end‐points of the path. Due to the fact that is skew‐symmetric, we have, for , Therefore, to construct energy‐preserving schemes, here we adopt the HBVMs discussed in , the main advantage of HBVMs is that it can precisely conserve the energy along the numerical solution.…”

“…As is well known, the value of Hamiltonian function is a constant along the solution of (3.1). According to [38,48,50], this can be easily verified by line integral associated with the vector field ∇H (Z) evaluated along the path defined by the solution Z (t) of (3.1), which equals the variation of H along the end-points of the path. Due to the fact that J is skew-symmetric, we have, for t 0 ≤ t ≤ t n ,…”

Line Integral Solution of Differential Problems

“…Relevant instances of line integral methods are provided by the energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) and the Energy and QUadratic Invariants Preserving (EQUIP) methods, providing efficient geometric integrators for Hamiltonian problems. It is worth mentioning that HBVMs can be easily adapted to efficiently handle Hamiltonian BVPs [19], whereas EQUIP methods can be also used for numerically solving Poisson problems. In this paper, we have also reviewed some active research trends, concerning the application of HBVMs for numerically solving constrained Hamiltonian problems, Hamiltonian PDEs, and highly-oscillatory problems.…”

“…Further generalizations, along several directions, have been considered in [19][20][21][22][23][24][25][26][27][28]: in particular, in [19] Hamiltonian boundary value problems have been considered, which are not covered in this review. The main reference on line integral methods is given by the monograph [1].…”

Line Integral Solution of Hamiltonian PDEs

“…with J = −J and H a scalar function (which we shall hereafter assume to be suitably regular), called the Hamiltonian or energy. Due to the skew-symmetry of J, this latter function turns out to be conserved along the solution of (1). In fact, one has: Hamiltonian problems are very important in the applications and, for this reason, their numerical simulation has been the subject of many researches: we refer the reader, e.g., to the monographs [6,19,53,62,67] and references therein.…”