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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 work we consider a mixed precision approach to accelerate the implemetation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a…
We present a non-canonically symplectic integration scheme tailored to numerically computing the post-Newtonian motion of a spinning black-hole binary. Using a splitting approach we combine the flows of orbital and spin contributions. In…
Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be…
In this paper, explicit stable integrators based on symplectic and contact geometries are proposed for a non-autonomous ordinarily differential equation (ODE) found in improving convergence rate of Nesterov's accelerated gradient method.…
Hamiltonian systems are one of the most important class of dynamical systems with a geometric structure called symplecticity and the numerical algorithms which can preserve such geometric structure are of interest. In this article we study…
Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general…
In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge--Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general…
Numerical integrators could be used to form interpolation conditions when training neural networks to approximate the vector field of an ordinary differential equation (ODE) from data. When numerical one-step schemes such as the Runge-Kutta…
Symplectic integrators with long-term preservation of integrals of motion are introduced for the guiding-center model of plasma particles in toroidal magnetic fields of general topology. An efficient transformation to canonical coordinates…
Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics, with the isospectral property often corresponding to mathematically or physically…
The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). The main…
We consider high order, implicit Runge-Kutta schemes to solve time-dependent stiff PDEs on dynamically adapted grids generated by multiresolution analysis for unsteady problems disclosing localized fronts. The multiresolution finite volume…
The main objective of this series of papers is to explore the entire landscape of numerical methods for fast nonlinear Fourier transformation (NFT) within the class of integrators known as the exponential integrators. In this paper, we…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…
This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions…
We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit-explicit Runge-Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in…
Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…
This article considers non-relativistic charged particle dynamics in both static and non-static electromagnetic fields, which are governed by nonseparable, possibly time-dependent Hamiltonians. For the first time, explicit symplectic…