Related papers: A classification of volume preserving generating f…
Regularizing volume preserving diffeomorphism (VPD) is equivalent to a long standing problem, namely regularizing Nambu-Poisson bracket. In this paper, as a first step to regularizing VPD, we find general complete independent basis of VPD…
This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…
Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for…
The continuous 1D defects of an isotropic homogeneous material in a flat 3D space are classified by the Volterra process construction method. We employ the same method to classify the continuous 2D defects of a vacuum in a 4D maximally…
The algebra of volume-preserving vector fields is considered. The potentials for that fields are introduced, and induced algebra of potentials is considered. It is shown, that this algebra fails to satisfy the Jacoby identity. Analogy with…
We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the…
Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic…
The spatial discretization of convective terms in compressible flow equations is studied from an abstract viewpoint, for finite-difference methods and finite-volume type formulations with cell-centered numerical fluxes. General conditions…
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…
Stationary and isotropic iteration stable random tessellations are considered, which can be constructed by a random process of cell division. The collection of maximal polytopes at a fixed time $t$ within a convex window $W\subset{\Bbb…
This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as…
Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems. In the mainstream of this research, we have defined a new family of symplectic integrators depending on a real…
Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial…
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of…
We give a possible extension for shears and overshears in the case of two non commutative (quaternionic) variables in relation with the associated vector fields and flows. We present a possible definition of volume preserving automorphisms,…
We present a new class of high-order variational integrators on Lie groups. We show that these integrators are symplectic, momentum preserving, and can be constructed to be of arbitrarily high-order, or can be made to converge…
We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. Variational integrators are an important class of geometric integrators. The general idea…