Time finite element methods: A unified framework for numerical discretizations of ODEs

On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems

“…Consequently, we notice that one needs only to compute once and for all matrix Σ (or factorize Σ −1 ), having the same size as that of the continuous problem. Secondly, in order to gain convergence for relatively large values of ωh, it is important to choose an appropriate starting value ψ 0 in (47). For this purpose, we use the solution of the associated homogeneous problem (14).…”

High order symplectic integrators based on continuous-stage Runge-Kutta-Nyström methods

“…In addition, continuous-stage approaches may promote the investigation of conjugate symplecticity of energy-preserving methods [16,17,32]. Besides, as shown in [30,34,35], some Galerkin variational methods can be interpreted as continuous-stage (P)RK methods, but they can not be clearly understood in the classical (P)RK framework. Therefore, the concept of continuous-stage methods provides us a larger realm for numerical discretization of differential equations and it opens a new insight for us in geometric integration.…”

Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation

“…For non-polynomial cases, a practical energy-preserving scheme is usually gained in the sense that the energy error remains bounded within machine precision, but one has to increase RK stages [2]. Recently, energy-preserving continuous stage RK (CSRK) methods have been attracting a lot of interest [21,30,33,38]. This kind of methods can eliminate the limit of HBVMs to cover non-polynomial Hamiltonian systems.…”