Related papers: Invariance correction to Grad's equations: Where t…

We make a brief historical review to the moment model reduction to the kinetic equations, particularly the Grad's moment method for Boltzmann equation. The focus is on the hyperbolicity of the reduced model, which is essential to the…

In the paper the possible approaches to the rigorous derivation of the Boltzmann kinetic equation with hard sphere collisions from underlying dynamics are considered. In particular, a formalism for the description of the evolution of…

We introduce a numerical method for solving Grad's moment equations or regularized moment equations for arbitrary order of moments. In our algorithm, we do not need explicitly the moment equations. As an instead, we directly start from the…

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high…

A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be…

This paper is to study the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the solution to the initial value problem is established for the Maxwellian molecules in a space of…

Despite the progress achieved by kinetic theory, the search of possible exact kinetic equations remains elusive to date. This concerns, specifically, the issue of the validity of the conjecture proposed by Grad (Grad, 1972) and developed in…

In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed…

A new approach of implementing initial and boundary conditions for the lattice Boltzmann method is presented. The new approach is based on an extended collision operator that uses the gradients of the fluid velocity. The numerical…

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type firstly proposed in \cite{Grad}. The Grad's moment method was originally…

In this paper, explicit method of constructing approximations (the Triangle Entropy Method) is developed for nonequilibrium problems. This method enables one to treat any complicated nonlinear functionals that fit best the physics of a…

This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to {\phi}-divergences and a corresponding…

We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses…

In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence…

One of the biggest challenges for simulating the Boltzmann equation is the evaluation of fivefold collision integral. Given the recent successes of deep learning and the availability of efficient tools, it is an obvious idea to try to…

One of the very few mathematically rigorous nonlinear model reduction methods is the restriction of a dynamical system to a low-dimensional, sufficiently smooth, attracting invariant manifold. Such manifolds are usually found using local…

The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such…

In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the…

Computational modeling and simulation of fluid-structure interactions constitute a fundamental cornerstone for advancing aerospace engineering endeavors. This paper addresses the notion and implementation of the immersed boundary method for…

In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient…