Related papers: Forward Symplectic Integrators for Solving Gravita…
Optimization tasks are crucial in statistical machine learning. Recently, there has been great interest in leveraging tools from dynamical systems to derive accelerated and robust optimization methods via suitable discretizations of…
By exploiting the error functions of explicit symplectic integrators for solving separable Hamiltonians, I show that it is possible to develop explicit, time-reversible symplectic integrators for solving non-separable Hamiltonians of the…
In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and…
In this note, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [Sakamoto-van~der Schaft, IEEE Transactions on Automatic Control, 2008]. Our algorithm…
In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is…
This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained…
We present an efficient and parsimonious algorithm to solve mixed initial/final-value problems. The algorithm optimally limits the memory storage and the computational time requirements: with respect to a simple forward integration, the…
The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). The main…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for…
Dynamic systems have a fundamental relevance in the description of physical phenomena. The search for more accurate and faster numerical integration methods for the resolution of such systems is, therefore, an important topic of research.…
We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We…
A new algorithm for numerical integration of the rigid-body equations of motion is proposed. The algorithm uses the leapfrog scheme and the quantities involved are angular velocities and orientational variables which can be expressed in…
This work considers the {\em gravitational} $N$-body problem and introduces global time-renormalization {\em functions} that allow the efficient numerical integration with fixed time-steps. First, a lower bound of the radius of convergence…
In this study, we present a simulation-based numerical method for solving a class of singularly perturbed second-order differential equations that come from a simplified biologically motivated model of human gait. Important physical factors…
Standard gradient-based iteration algorithms for optimization, such as gradient descent and its various proximal-based extensions to nonsmooth problems, are known to converge slowly for ill-conditioned problems, sometimes requiring many…
We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with…
The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of…
Magnetic quadrupoles are essential components of particle accelerators like the Large Hadron Collider. In order to study numerically the stability of the particle beam crossing a quadrupole, a large number of particle revolutions in the…
We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the…
Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for…