Related papers: Lie-Poisson methods for isospectral flows
A systematic digital-discrete method for obtaining continuous functions with smoothness to a certain order (C^(n)) from sample data is designed. This method is based on gradually varied functions and the classical finite difference method.…
Sufficient and necessary conditions are presented for the order-preservation of stochastic functional differential equations on $\R^d$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency…
Numerical simulation of compressible fluid flows is performed using the Euler equations. They include the scalar advection equation for the density, the vector advection equation for the velocity and a given pressure dependence on the…
We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian…
We show that the ideal (nondissipative) form of the dynamical equations for the Lipps-Hemler formulation of the anelastic fluid model follow as Euler-Poincar\'{e} equations, obtained from a constrained Hamilton's principle expressed in the…
In this paper we design discrete port-Hamiltonian systems systematically in two different ways, by applying discrete gradient methods and splitting methods respectively. The discrete port-Hamiltonian systems we get satisfy a discrete notion…
This paper concerns the well-posedness of subsonic Euler-Poisson flows in a convergent nozzle. Due to the geometry of the nozzle, we first introduce a coordinate transformation to prove the existence of radially symmetric subsonic solutions…
We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e…
In this paper we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual…
This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian…
We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on…
We consider the Euler equation of quasi-geostrophic fluids which is widely used in weather forecast. Our goal is to study explicit volume-preserving numerical methods for very long simulations on an energy and enstrophy preserving…
The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner's flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow…
It is well known that classical formulations resembling the Horn and Schunck model are still largely competitive due to the modern implementation practices. In most cases, these models outperform many modern flow estimation methods. In view…
A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a…
In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (i.e. truncated current…
Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for…
We analyze the canonical treatment of classical constrained mechanical systems formulated with a discrete time. We prove that under very general conditions, it is possible to introduce nonsingular canonical transformations that preserve the…
Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants.…
The recent interest in structure preserving stochastic Lagrangian and Hamiltonian systems raises questions regarding how such models are to be understood and the principles through which they are to be derived. By considering a…