Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. For example, it is well known that classical symplectic methods can only exactly preserve, at most, quadratic Hamiltonians. In this paper, a new family of methods, called "Hamiltonian Boundary Value Methods (HBVMs)", is introduced and analyzed. HBVMs are able to exactly preserve, in the discrete solution, Hamiltonian functions of polynomial type of arbitrarily high degree. These methods turn out to be symmetric, precisely A-stable, and can have arbitrarily high order. A few numerical tests confirm the theoretical results.Comment: 25 pages, 8 figures, revised versio

A note on the efficient implementation of Hamiltonian BVMs

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems, via their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.Comment: 10 pages, 4 figure

Enhanced HBVMs for the numerical solution of Hamiltonian problems with multiple invariants

Recently, the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs), has been proposed for the efficient solution of Hamiltonian problems, as well as for other types of conservative problems. In this paper, we report further advances concerning such methods, resulting in their enhanced version (Enhanced HBVMs, or EHBVMs). The basic theoretical results are sketched, along with a few numerical tests on a Hamiltonian problem, taken from the literature, possessing multiple invariants.Comment: 4 page

Blended General Linear Methods based on Boundary Value Methods in the GBDF family

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, order barriers for stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been made in order to derive GLMs with particular features, to be exploited for their efficient implementation. In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE-IVPs. In particular, this approach has been recently developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose theory is here completed and fully worked-out. Moreover, for each one of such methods, it is possible to define a corresponding Blended GLM which is equivalent to it from the point of view of the stability and order properties. These blended methods, in turn, allow the definition of efficient nonlinear splittings for solving the generated discrete problems. A few numerical tests, confirming the excellent potential of such blended methods, are also reported.Comment: 22 pages, 8 figure

Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed

Parallel Factorizations in Numerical Analysis

In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.Comment: 15 pages, 5 figure

Matrix stiffness affects endocytic uptake of MK2-inhibitor peptides.

In this study, the role of substrate stiffness on the endocytic uptake of a cell-penetrating peptide was investigated. The cell-penetrating peptide, an inhibitor of mitogen-activated protein kinase activated protein kinase II (MK2), enters a primary mesothelial cell line predominantly through caveolae. Using tissue culture polystyrene and polyacrylamide gels of varying stiffness for cell culture, and flow cytometry quantification and enzyme-linked immunoassays (ELISA) for uptake assays, we showed that the amount of uptake of the peptide is increased on soft substrates. Further, peptide uptake per cell increased at lower cell density. The improved uptake seen on soft substrates in vitro better correlates with in vivo functional studies where 10-100 µM concentrations of the MK2 inhibitor cell penetrating peptide demonstrated functional activity in several disease models. Additional characterization showed actin polymerization did not affect uptake, while microtubule polymerization had a profound effect on uptake. This work demonstrates that cell culture substrate stiffness can play a role in endocytic uptake, and may be an important consideration to improve correlations between in vitro and in vivo drug efficacy