Related papers: Geometric, Variational Discretization of Continuum…
We study a possibly integrable model of abelian gauge fields on a two-dimensional surface M, with volume form mu. It has the same phase space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M,…
In this study we have developed a flexible and efficient numerical scheme for the simulation of three-dimensional incompressible flows in spherical coordinates. The main idea, inspired by a similar strategy as (Verzicco, R., Orlandi, P.,…
Thermodynamically consistent models in continuum physics, i.e. models which satisfy the first and second laws of thermodynamics, may be expressed using the metriplectic formalism. In this work, we leverage the structures underlying this…
Lattice actions and amplitudes that perfectly mirror continuum physics are known as perfect discretizations. Such perfect discretizations naturally preserve the symmetries of the continuum. This is a key concern for general relativity,…
Discretization of general relativity is a promising route towards quantum gravity. Discrete geometries have a finite number of degrees of freedom and can mimic aspects of quantum geometry. However, selection of the correct discrete freedoms…
We develop a rigorous theory for a structure-preserving discretisation of the incompressible Euler and Navier--Stokes equations, based on discrete exterior calculus on prismatic Delaunay--Voronoi meshes over closed Riemannian manifolds. The…
We parameterize the phase space density by time dependent diffeomorphic, Poisson preserving transformations on phase space acting on a reference density solution. We can look at these as transformations which fix time on the extended space…
We review and discuss the role of diffeomorphism symmetry in quantum gravity models. Such models often involve a discretization of the space-time manifold as a regularization method. Generically this leads to a breaking of the symmetries to…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
In the pattern matching approach to imaging science, the process of ``metamorphosis'' is template matching with dynamical templates. Here, we recast the metamorphosis equations of into the Euler-Poincare variational framework of and show…
A numerical scheme is developed for systems of conservation laws on manifolds which arise in high speed aerodynamics and magneto-aerodynamics. The systems are presented in an arbitrary coordinate system on the manifold and involve source…
We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group of all smooth and orientation-preserving diffeomorphisms on the circle. These include…
In this review we discuss the interplay between discretization, constraint implementation, and diffeomorphism symmetry in Loop Quantum Gravity and Spin Foam models. To this end we review the Consistent Discretizations approach, which is an…
A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the…
We propose new formulations of geometric curvature flows -- referred to as \emph{dual formulations} -- that are equivalent to the original formulations but provide a novel framework for constructing linearly implicit and energy-stable…
Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows,…
Direct numerical simulation of microscale fluid--structure interactions in multicomponent and multiphase flows requires methods that can represent moving boundaries together with fields constrained to evolving interfaces. Diffuse-domain…
This paper proposes a novel first-order and a novel second-order fully discrete virtual element schemes based on the scalar auxiliary variable method for the three dimensional inductionless magnetohydrodynamics problem. The backward Eular…
This study proposes and analyses a novel higher-order, structure preserving discretization method for inviscid barotropic flows from a Lagrangian perspective. The method is built on a multisymplectic variational principle discretized over a…
In (J. Comput. Phys., 417, 109577, 2020) we introduced a space-time embedded-hybridizable discontinuous Galerkin method for the solution of the incompressible Navier-Stokes equations on time-dependent domains of which the motion of the…