Luigi Brugnano

A simple framework for the derivation and analysis of effective one-step methods for ODEs☆

In this paper, we provide a simple framework to derive and analyse a class of one-step methods that may be conceived as a generalization of the class of Gauss methods. The framework consists in coupling two simple tools: firstly a local Fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge–Kutta methods can be derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see [2] and references therein). A few numerical tests are also included, in order to confirm the effectiveness of the methods resulting from our analysis.

Iterative Solution of Piecewise Linear Systems

The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.

Convergence and stability of boundary value methods for ordinary differential equations

A usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula. If the formula has k steps, k values are needed to obtain the discrete solution. The continuous problem provides only the initial value. It is customary to impose the additional k - 1 conditions at the successive k - 1 initial points. However, the class of methods obtained in this way suffers from heavy limitations summarized by the two Dahlquist barriers. It is also possible to impose the additional conditions at different grid-points. For example, some conditions can be imposed at the initial points and the remaining ones at the final points. The obtained methods, called boundary value methods (BVMs), do not have barriers whatsoever. In this paper the question of convergence of BVMs is discussed, along with the linear stability theory. Some numerical examples on stiff test problems are also presented.

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

Abstract One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants; among them 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, we report the theoretical foundations which have led to the definition of the new family of methods, called Hamiltonian Boundary Value Methods (HBVMs). 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 and can have arbitrarily high order. A few numerical tests confirm the theoretical results.

Energy- and Quadratic Invariants--Preserving Integrators Based upon Gauss Collocation Formulae

We introduce a new family of symplectic integrators depending on a real parameter α. For α = 0, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of the internal stages. For any given non-null α, the corresponding method remains symplectic and has order 2s − 2: hence it may be interpreted as a O(h) (symplectic) perturbation of the Gauss method. Under suitable assumptions, we show that the parameter α may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order 2s as the generating Gauss formula.We introduce a new family of symplectic integrators for canonical Hamiltonian systems. Each method in the family depends on a real parameter

Iterative Solution of Piecewise Linear Systems and Applications to Flows in Porous Media

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Parallel factorizations for tridiagonal matrices

. When

The lack of continuity and the role of infinite and infinitesimal in numerical methods for ODEs: The case of symplecticity

\alpha=0

Hamiltonian BVMs (HBVMs): A Family of "Drift Free" Methods for Integrating polynomial Hamiltonian problems'

we obtain the classical Gauss collocation formula of order

Blended implementation of block implicit methods for ODEs

2s