Related papers: Structure preserving discretization: A Berezin-Toe…
Stability and convergence of full discretizations of various surface evolution equations are studied in this paper. The proposed discretization combines a higher-order evolving-surface finite element method (ESFEM) for space discretization…
In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by…
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
We develop a structure-preserving computational framework for optimal mixing control in incompressible flows. Our approach exactly conserves the continuous system's key invariants (mass and $L^2$-energy), while also maintaining discrete…
In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in local coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion…
In this article, we present the structure-preserving discretization of linear one-dimensional port-Hamiltonian (PH) systems of two conservation laws using discontinuous Galerkin (DG) methods. We recall the DG discretization procedure which…
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained…
In this thesis we study geometric structures from Poisson and generalized complex geometry with mild singular behavior using Lie algebroids. The process of lifting such structures to their Lie algebroid version makes them less singular, as…
In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate sub-problems. The coupled…
In the study of open quantum systems, one commonly describes the evolution of a system of interest through reduced dynamics, obtained by treating the environment indirectly rather than as a part of the full model. This thesis presents an…
A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce-Pillay, is formulated in terms of a relative notion of prolongation for Kolchin-closed…
This paper studies the construction of dynamic symbolic abstractions for nonlinear control systems via dynamic quantization. Since computational complexity is a fundamental problem in the use of discrete abstractions, a dynamic quantizer…
In this paper, we establish new quantitative convergence bounds for a class of functional autoregressive models in weighted total variation metrics. To derive our results, we show that under mild assumptions, explicit minorization and…
In this work, we conduct a systematic study of Hamiltonian and quasi-Hamiltonian systems within the framework of nondecomposable generalized Poisson geometry. Our focus lies on the interplay between the algebraic structure of…
In this work we present a novel structure-preserving scheme for the discretization of the Godunov-Peshkov-Romenski (GPR) model of continuum mechanics written in Lagrangian form. This model admits an extra conservation law for the total…
Highly accurate simulations of problems including second derivatives on complex geometries are of primary interest in academia and industry. Consider for example the Navier-Stokes equations or wave propagation problems of acoustic or…
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…
In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is…
We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem…