Related papers: Locally exact modifications of numerical integrato…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they…
Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved…
Hamiltonian systems are known to conserve the Hamiltonian function, which describes the energy evolution over time. Obtaining a numerical spatio-temporal scheme that accurately preserves the discretized Hamiltonian function is often a…
The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified…
A general framework for the numerical approximation of evolution problems is presented that allows to preserve exactly an underlying Hamiltonian- or gradient structure. The approach relies on rewriting the evolution problem in a particular…
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…
We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist Hamiltonians introduced for quasi-incompressible…
We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic, momentum preserving, and can be constructed to be of arbitrarily high-order, or can be made to converge…
Hamiltonian Monte Carlo is a prominent Markov Chain Monte Carlo algorithm, which employs symplectic integrators to sample from high dimensional target distributions in many applications, such as statistical mechanics, Bayesian statistics…
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…
In this paper, we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods. It is shown that these novel methods can be of…
A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic…
As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper,…
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are…
This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference…
We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple $({\mathcal D}^*, \Pi, \mathcal{H})$, where ${\mathcal D}^*$ is the dual of the…
Conservative symmetric second-order one-step integrators are derived using the Discrete Multiplier Method for a family of vortex-blob models approximating the incompressible Euler's equations on the plane. Conservative properties and second…
We present an adaptation of the so-called structural method \cite{CMM23} for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem…
We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is…