Related papers: Volume-preserving exponential integrators
It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume…
In this paper, we consider exponential integrators for semilinear Poisson systems. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for…
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete…
The primary objective of this paper is to present a long-term numerical energy-preserving analysis of one-stage explicit symmetric and/or symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators for highly oscillatory Hamiltonian…
We prove that the recently developed semiexplicit symplectic integrators for non-separable Hamiltonian systems preserve any linear and quadratic invariants possessed by the Hamiltonian systems. This is in addition to being symmetric and…
In this paper, we develop a framework to construct energy-preserving methods for multi-components Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method. This leads to numerical schemes…
Hamiltonian systems of ordinary and partial differential equations are fundamental mathematical models spanning virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is…
In earlier work, Lomeli and Meiss used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the $\mathbb{R}^3$ case, the first to differ from the symplectic case, they…
This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show…
We construct high order symmetric volume-preserving methods for the relativistic dynamics of a charged particle by the splitting technique with processing. Via expanding the phase space to include time $t$, we give a more general…
We consider the numerical simulation of Hamiltonian systems of ordinary differential equations. Two features of Hamiltonian systems are that energy is conserved along trajectories and phase space volume is preserved by the flow. We want to…
In this paper, we focus on the construction of high order volume preserving in- tegrators for divergence-free vector fields: the monomial basis, the exponential basis and tensor product of the monomial and the exponential basis. We first…
This paper is concerned with geometric exponential energy-preserving integrators for solving charged-particle dynamics in a magnetic field from normal to strong regimes. We firstly formulate the scheme of the methods for the system in a…
In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schr\"{o}dinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
We study numerical integrators that contract phase space volume even when the ODE does so at an arbitrarily small rate. This is done by a splitting into two-dimensional contractive systems. We prove a sufficient condition for Runge-Kutta…
Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important…
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a…
In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned…
As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper,…