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This paper presents a new mathematical model of vehicular traffic, based on the methods of the generalized kinetic theory, in which the space of microscopic states (position and velocity) of the vehicles is genuinely discrete. While in the…
This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model…
Discrete gradients (DG) or more exactly discrete gradient methods are time integration schemes that are custom-built to preserve first integrals or Lyapunov functions of a given ordinary differential equation (ODE). In conservative…
Both discrete and continuum models have been widely used to study rapid granular flow, discrete model is accurate but computationally expensive, whereas continuum model is computationally efficient but its accuracy is doubtful in many…
This paper develops a class of robust weak Galerkin methods for the stationary incompressible convective Brinkman-Forchheimer equations. The methods adopt piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively for the…
Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian…
The kinetic Boltzmann equation models gas dynamics over a wide range of spatial and temporal scales. Simplified versions of the full Boltzmann collision operator, such as the classical Bhatnagar-Gross-Krook and the closely related…
Furihata and Matsuo proposed in 2010 an energy-conserving scheme for the Zakharov equations, as an application of the discrete variational derivative method (DVDM). This scheme is distinguished from conventional methods (in particular the…
The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…
We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the…
We propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and the flow…
Predicting the occurence of rare and extreme events in complex systems is a well-known problem in non-equilibrium physics. These events can have huge impacts on human societies. New approaches have emerged in the last ten years, which…
The paper proves existence of renormalized stationary solutions for a dense class of discrete velocity Boltzmann equations in the plane with given ingoing boundary values. The proof is based on the construction of a sequence of…
Electrons are the carriers of heat and electricity in materials, and exhibit abundant transport phenomena such as ballistic, diffusive, and hydrodynamic behaviors in systems with different sizes. The electron Boltzmann transport equation…
Discrete Boltzmann model (DBM) is a type of coarse-grained mesoscale kinetic model derived from the Boltzmann equation. Physically, it is roughly equivalent to a hydrodynamic model supplemented by a coarse-grained model for the relevant…
Dynamic Mode Decomposition (DMD) is a data-driven method for approximating the spatiotemporal modes of a system. The eigenvectors and eigenvalues of the system are approximated from a series of time-snapshots of the state variables. The…
A hybrid lattice Boltzmann method (LBM) for binary mixtures based on the free-energy approach is proposed. Non-ideal terms of the pressure tensor are included as a body force in the LBM kinetic equations, used to simulate the continuity and…
Safety-critical traffic simulation plays a crucial role in evaluating autonomous driving systems under rare and challenging scenarios. However, existing approaches often generate unrealistic scenarios due to insufficient consideration of…
In this paper, a unified gas kinetic scheme with adaptive velocity space (AUGKS) for multiscale flow transport will be developed. In near-equilibrium flow regions, particle distribution function is close to the Chapman-Enskog expansion and…
The sequential nature of autoregressive next-token prediction imposes a fundamental speed limit on large language models. While continuous flow models offer a path to parallel generation, they traditionally demand expensive iterative…