Related papers: Numerically modelling semidirect product geodesics

We show that the symplectic reduction of the dynamics of $N$ point vortices on the plane by the special Euclidean group $\mathsf{SE}(2)$ yields a Lie--Poisson equation for relative configurations of the vortices. Specifically, we combine…

We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid…

This paper considers weak Galerkin finite element approximations for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k \ (k\geq 1)$ for the stress approximation, degree $k+1$ for…

This paper deals with the numerical resolution of kinetic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. We focus on the kinetic model considered in [18, 17] where alignment is…

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class…

Arnold showed that the Euler equations of an ideal fluid describe geodesics on the Lie algebra of incompressible vector fields. We generalize this to fluids with dissipation and Gaussian random forcing. The dynamics is determined by the…

For a class of flows on polytopes, including many examples from Evolutionary Game Theory, we describe a piecewise linear model which encapsulates the asymptotic dynamics along the heteroclinic network formed out of the polytope's vertexes…

First of all, we recall the well known notion of semidirect product both for classical algebraic structures (like groups and rings) and for more recent ones (digroups, left skew braces, heaps, trusses). Then we analyse the concept of…

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal…

We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $\theta$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and…

This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of…

We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group $G$-action has been considered on a strongly monotone skew-product semiflow. Here we relax the…

We present and analyze a structure-preserving method for the approximation of solutions to nonlinear cross-diffusion systems, which combines a Local Discontinuous Galerkin spatial discretization with the backward Euler time-stepping scheme.…

We analyze a nonlocal coupled system arising as the Euler--Lagrange equations of an energy functional involving regional fractional Laplacians of orders $s_1$ and $s_2$ ($ 0 < s_1,s_2 < 1$), each acting on a separate disjoint domain and…

The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using…

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i)…

We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with…

We propose and rigorously analyse semi- and fully discrete discontinuous Galerkin methods for an initial and boundary value problem describing inertial viscoelasticity in terms of elastic and viscoelastic stress components, and with mixed…

We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In…

The equations of motion of a charged ideal fluid, respectively the superconductivity equation (both in a given magnetic field) are showed to be geodesic equations on a general, respectively central extension of the group of volume…