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Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For…
We develop a semi-discrete optimal transport scheme for the compressible semi-geostrophic equations, a system that plays an important role in modelling large-scale atmospheric dynamics and frontogenesis. Unlike the incompressible case, the…
The subject of this paper is the evolution of the concept of information processing in regular structures based on multi-level processing in nested cellular automata. The essence of the proposed model is a discrete space-time containing…
We study higher-order space-time variational discretisations for modeling complex processes in porous media that include fluid and structure interactions which are of fundamental importance in many engineering fields with applications in…
Cellular automata are fully-discrete, spatially-extended dynamical systems that evolve by simultaneously applying a local update function. Despite their simplicity, the induced global dynamic produces a stunning array of richly-structured,…
In cellular automata with memory, the unchanged maps of the conventional cellular automata are applied to cells endowed with memory of their past states in some specified interval. We implement Rule 30 automata with a majority memory and…
We present a novel spatial discretization for the Cahn-Hilliard equation including transport. The method is given by a mixed discretization for the two elliptic operators, with the phase field and chemical potential discretized in…
Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic…
We perform numerical analysis of a nonlinear gradient flow, which can be regarded as a parabolic minimal surface problem or a regularised total variation flow, using the gradient discretisation method (GDM). GDM is a unified convergence…
Classical cellular automata represent a class of explicit discrete spacetime lattice models in which complex large-scale phenomena emerge from simple deterministic rules. With the goal to uncover different physically distinct classes of…
Two identical 1D autocatalytic systems with Gray--Scott kinetics--driven towards convectively unstable regimes and submitted to independent spatiotemporal Gaussian white noises--are coupled unidirectionally, but otherwise linearly.…
We introduce a novel optimization problem formulation that departs from the conventional way of minimizing machine learning model loss as a black-box function. Unlike traditional formulations, the proposed approach explicitly incorporates…
This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume discretizations. The proposed…
Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result,…
We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program…
We study discrete dynamical systems through the topological concepts of limit set, which consists of all points that can be reached arbitrarily late, and asymptotic set, which consists of all adhering values of orbits. In particular, we…
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
Various biological cells secrete diffusing chemical compounds into their environment for communication purposes. Secretion usually takes place over the cell membrane in a spatially heterogeneous manner. Mathematical models of these…
The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes whose goals are to preserve, at the discrete…
A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed…