Related papers: A gradient flow approach to linear Boltzmann equat…
We briefly introduce several fundamental problems that cause the creation of Discrete Boltzmann modeling and analysis Method(DBM), corresponding solutions, the relationship and difference between DBM and traditional fluid modeling and other…
In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…
We prove that the solution of certain linear stochastic differential equations in Hilbert spaces, namely those with bounded operators as well as the conservative stochastic Schr\"odinger equations, can be obtained - along the lines of the…
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…
We consider a gradient flow modeling the epitaxial growth of thin films with slope selection. The surface height profile satisfies a nonlinear diffusion equation with biharmonic dissipation. We establish optimal local and global…
A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics, and also as the gradient flow of a second-order information functional with respect to the…
To simulate highly-resolved flow fields, we extend the Quantum Lattice Boltzmann Method (QLBM) to be able to simulate multiple time steps without state extraction or reinitialization. We adjust and extend given QLBM approaches from the…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
Diffusion and flow-based models have become the state of the art for generative AI across a wide range of data modalities, including images, videos, shapes, molecules, music, and more. This tutorial provides a self-contained introduction to…
In this paper, computations of transient, incompressible, turbulent, plane jets using the discrete lattice BGK Boltzmann equation are reported. A priori derivation of the discrete lattice BGK Boltzmann equation with a spatially and…
The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar…
This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise…
We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order…
The gradient expansion is the fundamental organising principle underlying relativistic hydrodynamics, yet understanding its convergence properties for general nonlinear flows has posed a major challenge. We introduce a simple method to…
In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo…
Motivated by the recent work of Chigusa, Moroi, and Shoji, we propose a new simple gradient flow equation to derive the bounce solution which contributes to the decay of the false vacuum. Our discussion utilizes the discussion of Coleman,…
We derive Lattice Boltzmann (LBM) schemes to solve the Linearized Euler Equations in 1D, 2D, and 3D with the future goal of coupling them to an LBM scheme for Navier Stokes Equations and an Finite Volume scheme for Linearized Euler…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
Diffusion and flow-based generative models have achieved remarkable success in domains such as image synthesis, video generation, and natural language modeling. In this work, we extend these advances to weight space learning by leveraging…
Tracer diffusion in a granular gas in simple shear flow is analyzed. The analysis is made from a perturbation solution of the Boltzmann kinetic equation through first order in the gradient of the mole fraction of tracer particles. The…