Related papers: Variational Structures in Cochain Projection Based…
A variational integrator for ideal magnetohydrodynamics is derived by applying a discrete action principle to a formal Lagrangian. Discrete exterior calculus is used for the discretisation of the field variables in order to preserve their…
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in…
This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element…
We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete…
A class of high-order canonical symplectic structure-preserving geometric algorithms are developed for high-quality simulations of the quantized Dirac-Maxwell theory based strong-field quantum electrodynamics (SFQED) and relativistic…
A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property…
We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on…
We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the…
In this paper, we develop variational integrators for the nonequilibrium thermodynamics of simple closed systems. These integrators are obtained by a discretization of the Lagrangian variational formulation of nonequilibrium thermodynamics…
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for…
This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration…
Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader field of…
We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on…
We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The…
A variational method is used to derive a self-consistent macro-particle model for relativistic electromagnetic kinetic plasma simulations. Extending earlier work [E. G. Evstatiev and B. A. Shadwick, J. Comput. Phys., vol. 245, pp. 376-398,…
This article studies structure-preserving discretizations of Hilbert complexes with nonconforming spaces that rely on projections onto an underlying conforming subcomplex. This approach follows the conforming/nonconforming Galerkin (CONGA)…
We develop a semi-discrete version of discrete variational mechanics with applications to numerical integration of classical field theories. The geometric preservation properties are studied.
A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the…
We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the…