The aim of this study will be to design Parallel solver (PS) for oscillatory stiff systems ordinary differential equations (ODEs). PS constructed via a type specially transformed exponentially fitted multinomial approximant in accordance with the behaviour solution. method interpolation and collocation utilized. principal local truncation errors used derive suitable step size decide error toler...

We consider many-body problems in classical mechanics where a wide range of time scales limits what can be computed. We apply the method of optimal prediction to obtain equations that are easier to solve numerically. We demonstrate by examples that optimal prediction can reduce the amount of computation needed to obtain a solution by several orders of magnitude.

At the example of Hamiltonian differential equations, geometric properties of the flow are discussed that are only preserved by special numerical integrators (such as symplectic and/or symmetric methods). In the ‘non-stiff’ situation the long-time behaviour of these methods is well-understood and can be explained with the help of a backward error analysis. In the highly oscillatory (‘stiff’) ca...

We investigate numerical integration of ordinary differential equations (ODEs) for Hamiltonian Monte Carlo (HMC). High-quality integration is crucial for designing efficient and effective proposals for HMC. While the standard method is leapfrog (Störmer-Verlet) integration, we propose the use of an exponential integrator, which is robust to stiff ODEs with highly-oscillatory components. This os...

Geometric Numerical Integration is a subfield of the numerical treatment of differential equations. It deals with the design and analysis of algorithms that preserve the structure of the analytic flow. The present review discusses numerical integrators, which nearly preserve the energy of Hamiltonian systems over long times. Backward error analysis gives important insight in the situation, wher...

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed.Themethodswhich can be applied in predictor-corrector formare implemented in block formas simultaneous numerical integrators over nonove...

We present two families of explicit and implicit BDF formulas, capable of the exact integration (with only round-off errors) of differential equations which solutions belong to the space generated by the linear combinations of exponential of matrices, products of the exponentials by polynomials and products of those matrices by ordinary polynomials. Those methods are suitable for stiff and high...

In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highl...