The purpose of this work is to introduce a new idea how avoid the factorization large matrices during solution stiff systems ODEs. Starting from general form an explicit linear multistep method we suggest adaptively choose its coefficients on each integration step in order minimize norm residual implicit BDF formula. Thereby reduce number unknowns n O(1), where dimension ODE system. We call typ...

Certain symmetric linear multistep methods have an excellent long-time behavior when applied to second order Hamiltonian systems with or without constraints. For high accuracy computations round-off can be the dominating source of errors. This article shows how symmetric multistep methods should be implemented, so that round-off errors are minimized and propagate like a random walk.

This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analys...