Related papers: Symplectic Runge-Kutta Methods for Hamiltonian Sys…
We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the…
In this work, we develop a class of up to third-order energy-stable schemes for the Cahn--Hilliard equation. Building on Lawson's integrating factor Runge--Kutta method, which is widely used for stiff semilinear equations, we discuss its…
In this work, a kernel-based surrogate for integrating Hamiltonian dynamics that is symplectic by construction and tailored to large prediction horizons is proposed. The method learns a scalar potential whose gradient enters a…
We construct explicit integrators of arbitrary even orders of accuracy for massive point vortex dynamics in binary mixture of Bose--Einstein condensates proposed by Richaud et al. The integrators are symplectic and preserve the angular…
In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for…
Symplectic integrators offer many advantages for the numerical solution of Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two of the central Hamiltonian systems encountered in…
This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for…
Inductive bias refers to restrictions on the hypothesis class that enable a learning method to generalize effectively from limited data. A canonical example in control is linearity, which underpins low sample-complexity guarantees for…
In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such novel strategy is based on the newly developed exponential scalar…
We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is…
We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian…
In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced…
A recently developed high-order implicit shock tracking (HOIST) framework for resolving discontinuous solutions of inviscid, steady conservation laws [41, 43] is extended to the unsteady case. Central to the framework is an optimization…
Fractional-step methods are a popular and powerful divide-and-conquer approach for the numerical solution of differential equations. When the integrators of the fractional steps are Runge--Kutta methods, such methods can be written as…
We show that the massive noncommutative U(1) theory is embedded in a gauge theory using an alternative systematic way, which is based on the symplectic framework. The embedded Hamiltonian density is obtained after a finite number of steps…
In this paper we give an overview of Implicit-Explicit Runge-Kutta schemes applied to hyperbolic systems with stiff relaxation. In particular, we focus on some recent results on the uniform accuracy for hyperbolic systems with stiff…
In this paper, we present algebraic tools to obtain normal forms of $\omega$-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The…
We detect, by using symplectic topology, invariant measures with large rotation vectors for a class of Hamiltonian flows.
We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete…
This paper presents a method for learning Hamiltonian dynamics from a limited set of data points. The Hamiltonian vector field is found by regularized optimization over a reproducing kernel Hilbert space of vector fields that are inherently…