This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce simplicial Dirac structures as discrete analogues of the Stokes-Dirac structure and demonstrate that they provide a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, we preserve a number of important topological and geometrical properties of the system.

The underlying structure of port-Hamiltonian systems considered in this paper is a Stokes-Dirac structure [1] and as such is defined on a certain space of differential forms on a smooth finite-dimensional orientable manifold with a boundary. The Stokes-Dirac structure generalizes the framework of the Poisson and symplectic structures by providing a theoretical account that permits the inclusion of varying boundary variables in the boundary problem for partial differential equations. From an interconnection and control viewpoint, such a treatment of boundary conditions is essential for the incorporation of energy exchange through the boundary, since in many applications the interconnection with the environment takes place precisely through the boundary. For numerical integration, simulation and control synthesis, it is of paramount interest to have finite approximations that can be interconnected to one another or via the boundary coupled to the other systems, be they finite- or infinite-dimensional.

Abstract Reaction-diffusion systems model the evolution of the constituents distributed in space under the influence of chemical reactions and diffusion. These systems arise naturally in chemistry, but can also be used to model dynamical processes beyond the realm of chemistry such as in biology, ecology, geology, and physics. In this paper, by adopting the viewpoint of port-based modeling, we cast reaction-diffusion systems into the port-Hamiltonian framework. Aside from offering conceptually a clear geometric interpretation formalized by a Stokes-Dirac structure, a port-Hamiltonian perspective allows to treat these dissipative systems as interconnected and thus makes their analysis, both quantitative and qualitative, more accessible from a modern dynamical systems and control theory point of view. This modeling approach permits us to draw immediately some conclusions regarding passivity and stability of reaction-diffusion systems. Furthermore, by adopting a discrete differential geometry-based approach and discretizing the reaction-diffusion system in the port-Hamiltonian form, apart from preserving a geometric structure, a compartmental model analogous to the standard one is obtained.

Reaction-diffusion systems model the evolution of the constituents distributed in space under the influence of chemical reactions and diffusion [6], [10]. These systems arise naturally in chemistry [5], but can also be used to model dynamical processes beyond the realm of chemistry such as biology, ecology, geology, and physics. In this paper, by adopting the viewpoint of port-controlled Hamiltonian systems [7] we cast reaction-diffusion systems into the portHamiltonian framework. Aside from offering conceptually a clear geometric interpretation formalized by a Stokes-Dirac structure [8], a port-Hamiltonian perspective allows to treat these dissipative systems as interconnected and thus makes their analysis, both quantitative and qualitative, more accessible from a modern dynamical systems and control theory point of view. This modeling approach permits us to draw immediately some conclusions regarding passivity and stability of reaction-diffusion systems. It is well-known that adding diffusion to the reaction system can generate behaviors absent in the ode case. This primarily pertains to the problem of diffusion-driven instability which constitutes the basis of Turing’s mechanism for pattern formation [11], [5]. Here the treatment of reaction-diffusion systems as dissipative distributed portHamiltonian systems could prove to be instrumental in supply of the results on absorbing sets, the existence of the maximal attractor and stability analysis. Furthermore, by adopting a discrete differential geometrybased approach [9] and discretizing the reaction-diffusion system in port-Hamiltonian form, apart from preserving a geometric structure, a compartmental model analogous to the standard one [1], [2] is obtained

This paper considers the application of extrapolation techniques in finding approximate solutions of some optimization problems with constraints defined by the Robin boundary problem for the Laplace equation. When applied extrapolation techniques produce very accurate solutions of the boundary problems on relatively coarse meshes, but this paper demonstrates that this is not a real restriction when dealing with optimization problems. Producing a solution of continuous problem by polynomial extrapolation based on the low-order discrete problem solutions significantly reduces both computational time and memory. The present paper illustrates this approach using finite-difference and finite-element methods, and finally makes a brief remark about some tacit engineering assumptions regarding numerical solutions of conductive media problems by construction of equivalent resistor networks.

This paper presents a method to find a solution of a two-dimensional field problem using finite-difference approach. The finite-difference technique has a slow convergence, but it is simple to program, and has a clear geometric interpretation. In this paper a thin conducting sheet was replaced by a mesh, with variable number of nodes. The potential of an arbitrary point of the sheet was approximated by a function having as arguments the order of the mesh, represented by the number of the nodes, and three unknown parameters. The problem was subsequently solved for several meshes with low number of nods, and the values of the parameters were found by a fitting procedure. The obtained approximating function predicted well the solutions for higher order meshes. There is a possibility that it can also predict the exact solution of the continuous problem seen as an infinite- order mesh for well-posed cases.