Related papers: High order three part split symplectic integrators…
Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious…
In this methods paper, we show how to tridia\-go\-nalize two families of bosonic multimode systems: optomechanical and Bose-Hubbard hamiltonians. Using tools from number theory, we devise a rendering of these systems in the form of exact $D…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
We consider a class of Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part, and we analyze their numerical discretizations by symplectic methods when the initial value is small in Sobolev norms.…
We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE…
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for…
Hamiltonian systems of ordinary and partial differential equations are fundamental mathematical models spanning virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is…
The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant…
A class of Hamiltonian stochastic differential equations with multiplicative L\'{e}vy noise in the sense of Marcus, and the construction and numerical implementation methods of symplectic Euler scheme, are considered. A general symplectic…
We present a brief tutorial on the nuts and bolts computation of a multisymplectic particle-in-cell algorithm using the discretized Lagrangian approach. This approach, originated by Marsden, Shadwick, and others, brings the benefits of…
We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps…
We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate…
Two specialized algorithms for the numerical integration of the equations of motion of a Brownian walker obeying detailed balance are introduced. The algorithms become symplectic in the appropriate limits, and reproduce the equilibrium…
Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually…
We present a method for constructing numerical schemes with up to 3rd strong convergence order for solution of a class of stochastic differential equations, including equations of the Langevin type. The construction proceeds in two stages.…
In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes…
In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The…
This paper focuses on the construction and analysis of explicit numerical methods of high dimensional stochastic nonlinear Schrodinger equations (SNLSEs). We first prove that the classical explicit numerical methods are unstable and suffer…
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…