Related papers: Quasi symplectic integrators for stochastic differ…
Symplectic integrators are widely used for the study of planetary dynamics and other $N$-body problems. In a study of the outer Solar system, we demonstrate that individual symplectic integrations can yield biased errors in the semi-major…
Dependable numerical results from long-time simulations require stable numerical integration schemes. For Hamiltonian systems, this is achieved with symplectic integrators, which conserve the symplectic condition and exactly solve for the…
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of…
In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall…
Simulation of many-particle system evolution by molecular dynamics takes to decrease integration step to provide numerical scheme stability on the sufficiently large time interval. It leads to a significant increase of the volume of…
The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of…
Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. For the first time, the article proposes for arbitrary Hamiltonians similar integrators,…
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…
Solving quaternion kinematical differential equations is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of…
A parallel implementation of coupled spin-lattice dynamics in the LAMMPS molecular dynamics package is presented. The equations of motion for both spin only and coupled spin-lattice dynamics are first reviewed, including a detailed account…
We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent…
Symplectic integrators are widely used for long-term integration of conservative astrophysical problems due to their ability to preserve the constants of motion; however, they cannot in general be applied in the presence of nonconservative…
Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Lagrangian and Hamiltonian systems from the variational framework introduced in Wibisono et al. In this…
Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation…
We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in…
In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are…
We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the…
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…
Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly…
While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we…