Related papers: Multi-symplectic Preserving Integrator for the Sch…
This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving…
The multi-symplectic form for Hamiltonian PDEs leads to a general framework for geometric numerical schemes that preserve a discrete version of the conservation of symplecticity. The cases for systems or PDEs with dissipation terms has…
Using the maximal Lie algebra of point symmetries of a system of nonlinear equations used in geophysical fluid dynamics, two conservation laws are found in addition to the conservation of energy.
A new methodology is developed to integrate numerically the equations of motion for classical many-body systems in molecular dynamics simulations. Its distinguishable feature is the possibility to preserve, independently on the size of the…
In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the…
For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each…
Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and…
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…
An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon-matter interactions described by the Schr\"odinger-Maxwell equations. The algorithms preserve the symplectic…
We propose explicit symplectic integrators of molecular dynamics (MD) algorithms for rigid-body molecules in the canonical and isothermal-isobaric ensembles. We also present a symplectic algorithm in the constant normal pressure and lateral…
Structure-preserving geometric algorithm for the Vlasov-Maxwell (VM) equations is currently an active research topic. We show that spatially-discretized Hamiltonian systems for the VM equations admit a local energy conservation law in…
This paper presents applications of weighted meshless scheme for conservation laws to the Euler equations and the equations of ideal magnetohydrodynamics. The divergence constraint of the latter is maintained to the truncation error by a…
In this paper, a novel high-order, mass and energy-conserving scheme is proposed for the regularized logarithmic Schr\"{o}dinger equation(RLogSE). Based on the idea of the supplementary variable method (SVM), we firstly reformulate the…
Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode…
This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the…
Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, and the nonlinear Schr\"{o}dinger equations, the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the…
We propose new algorithms for numerical integration of the equations of motion for classical spin systems with fixed spatial site positions. The algorithms are derived on the basis of a mid-point scheme in conjunction with the multiple time…
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation…
Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of…
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,…