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...

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...

In this paper, we will present a review of the multistep collocation method for Delay Volterra Integral Equations (DVIEs) from [1] and then, we study the superconvergence analysis of the multistep collocation method for DVIEs. Some numerical examples are given to confirm our theoretical results.

Implicit-explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fast-wave, slow-wave problems in which the fast wave is physically insignificant and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on 5 Adams methods or on backward differen...

We approximate the solution of initial boundary value problems for equations of the form Au′(t) = B(t, u(t)), t ∈ [0, t?]. A is a linear, selfadjoint, positive definite operator on a Hilbert space (H, (·, ·)) and B is a possibly nonlinear operator. We discretize in space by finite element methods and for the time discretization we use explicit linear multistep schemes. We derive optimal order e...

Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation. Dedicated to Professor Vidar Thomée on the occasion of his 65 birthday, August 20, 1998

Implicit–explicit multistep methods for nonlinear parabolic equations were recently analyzed in [2, 3, 1]. In these papers the linear operator of the equation is assumed time-independent, self-adjoint and positive definite; then, the linear part is discretized implicitly and the remaining part explicitly. Here we slightly relax the hypotheses on the linear operator by allowing part of it to be ...