Efficient implementation of geometric integrators for separable Hamiltonian problems

We here investigate the efficient implementation of the energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) recently introduced for the numerical solution of Hamiltonian problems. In this note, we describe an iterative procedure, based on a triangular splitting, for solving the generated discrete problems, when the problem at hand is separable.Comment: 4 page

The Observability Radius of Networks

This paper studies the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix, and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights, with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks.Comment: 8 pages, 3 figure

The Observability Radius of Networks

This paper studies the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks

Energy conservation issues in the numerical solution of the semilinear wave equation

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.Comment: 41 pages, 11 figur

Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation

Conservation laws are among the most fundamental geometric properties of a given partial differential equation. However, standard finite difference approximations rarely preserve more than a single conservation law. A novel symbolic-numerical approach, introduced in [1], exploits the fact that divergences belong to the kernel of the Euler operator to construct schemes that preserve multiple conservation laws. However, this approach is limited by the complexity of the symbolic computations, whose cost is high even when the nonlinearity in the PDE is only quadratic. Some key simplifications, making the symbolic computations tractable, have been introduced in [2]. We apply this simplified strategy to the modified Korteweg-de Vries equation, having a cubic nonlinearity, to construct new bespoke finite-difference schemes that preserve the local conservation laws of the mass and of the energy

Recent Advances in the Numerical Solution of Hamiltonian Partial Differential Equations

In this paper, we study recent results in the numerical solution of Hamiltonian partial differential equations (PDEs), by means of energy-conserving methods in the class of Line Integral Methods, in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). We show that the use of energy-conserving methods, able to conserve a discrete counterpart of the Hamiltonian functional (which derives from a proper space semi-discretization), confers more robustness to the numerical solution of such problems

Solving the nonlinear Schrödinger equation using energy conserving Hamiltonian Boundary Value Methods

In this paper we study the use of energy-conserving methods, in the class of Hamiltonian Boundary Value Methods, for the numerical solution of the nonlinear Schrödinger equation

Deformation mechanisms in a continental rift up to mantle exhumation. Field evidence from the western Betics, Spain

International audienceThe identification of the structures and deformation patterns in magma-poor continental rifted margins is essential to characterize the processes of continental lithosphere necking. Brittle faults, often termed mantle detachments, are believed to play an essential role in the rifting processes that lead to mantle exhumation. However, ductile shear zones in the deep crust and mantle are rarely identified and their mechanical role remains to be established. The western Betics (Southern Spain) provides an exceptional exposure of a strongly thinned continental lithosphere, formed in a supra-subduction setting during Oligocene-Lower Miocene. A full section of the entire crust and the upper part of the mantle is investigated. Variations in crustal thickness are used to quantify crustal stretching that may reach values larger than 2000% where the ductile crust almost disappear, defining a stage of hyper-stretching. Opposite senses of shear top-to-W and top-to-E are observed in two extensional shear zones located close to the crust-mantle boundary and along the brittle-ductile transition in the crust, respectively. At locations where the ductile crust almost disappears, concordant top-to-E-NE senses of shear are observed in both upper crust and serpentinized mantle. Late high-angle normal faults with ages of ca. 21 Ma or older (40Ar/39Ar on white mica) crosscut the previously hyper-stretched domain, involving both crust and mantle in tilted blocks. The western Betics exemplifies, probably better than any previous field example, the changes in deformation processes that accommodate the progressive necking of a continental lithosphere. Three successive steps can be identified: i/a mid-crustal shear zone and a crust-mantle shear zone, acting synchronously but with opposite senses of shear, accommodate ductile crust thinning and ascent of subcontinental mantle; ii/hyper-stretching localizes in the neck, leading to an almost disappearance of the ductile crust and bringing the upper crust in contact with the subcontinental mantle, each of them with their already acquired opposite senses of shear; and iii/high-angle normal faulting, cutting through the Moho, with related block tilting, ends the full exhumation of the mantle in the zone of localized stretching. The presence of a high strength sub-Moho mantle is responsible for the change in sense of shear with depth. Whereas mantle exhumation in the western Betics occurred in a backarc setting, this deformation pattern controlled by a high-strength layer at the top of the lithosphere mantle makes it directly comparable to most passive margins whose formation lead to mantle exhumation. This unique field analogue has therefore a strong potential for the seismic interpretation of the so-called “hyper-extended margins”

A New Technique for Preserving Conservation Laws

This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes of order two and four that preserve three conservation laws. We show that the new technique is practicable for PDEs with three dependent variables, introducing as an example new families of second-order schemes for the potential Kadomtsev–Petviashvili equation