相关论文: Exponential integrators preserving first integrals…

The numerical integration of the Benjamin and Benjamin--Ono equations are considered. They are non-local partial differential equations involving the Hilbert transform, and due to this, so far quite few structure-preserving integrators have…

In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of…

This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for…

In this work we derive and analyze variational integrators of higher order for the structure-preserving simulation of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules…

Coarse-grained models that preserve hydrodynamics provide a natural approach to study collective properties of soft-matter systems. Here, we demonstrate that commonly used integration schemes in dissipative particle dynamics give rise to…

Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…

This paper introduces a robust reformulation of the incompressible Navier-Stokes equations, establishing a foundational framework for designing efficient, structure-preserving algorithms that strictly conserve the original energy…

A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied to the nonlinear Schroedinger equation,…

We study random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of systems, we establish an invariance principle stating that if all Lyapunov exponents vanish, then there exists a measurable family of…

This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the…

A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation…

In this paper, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a…

This paper introduces a family of entropy-conserving finite-difference discretizations for the compressible flow equations. In addition to conserving the primary quantities of mass, momentum, and total energy, the methods also preserve…

In this paper, we develop a structure-preserving discretization of the Lagrangian framework for electromagnetism, combining techniques from variational integrators and discrete differential forms. This leads to a general family of…

We introduce exponential numerical integration methods for stiff stochastic dynamical systems of the form $d\mathbf{z}_t = L(t)\mathbf{z}_tdt + \mathbf{f}(t)dt + Q(t)d\mathbf{W}_t$. We consider the setting of time-varying operators $L(t),…

In this paper, we develop efficient and accurate evaluation for the Lyapunov operator function $\varphi_l(\mathcal{L}_A)[Q],$ where $\varphi_l(\cdot)$ is the function related to the exponential, $\mathcal{L}_A$ is a Lyapunov operator and…

We study the problem of distance-based formation shape control for autonomous agents with double-integrator dynamics. Our considerations are focused on exponential stability properties. For second-order formation systems under the standard…

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…

Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general…

Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying…