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We consider here time-dependent three-dimensional stratified geophysical water flows of finite depth over a variable bottom with a free surface and an interface (separating two layers of constant and different densities). Under the…
In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…
In this work, we develop variational formulations of Petrov-Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann-Liouville or Caputo derivative of order $\alpha\in(3/2, 2)$ in the leading term and…
This paper investigates quenching solutions of an one-dimensional, two-sided Riemann-Liouville fractional order convection-diffusion problem. Fractional order spatial derivatives are discretized using weighted averaging approximations in…
The lattice Boltzmann (LB) equation with a third-order scheme can be regarded as a unified and self-consistent framework of the pseudopotential LB model for multiphase flows. In this work, we theoretically analyze pseudopotential LB…
In this paper, we propose a new numerical scheme for a spatially discrete model of constrained total variation flows, which are total variation flows whose values are constrained in a Riemannian manifold. The difficulty of this problem is…
A fully discrete finite element method, based on a new weak formulation and a new time-stepping scheme, is proposed for the surface diffusion flow of closed curves in the two-dimensional plane. It is proved that the proposed method can…
This work describes three diffuse-interface methods for the simulation of immiscible, compressible multiphase fluid flows and elastic-plastic deformation in solids. The first method is the localized-artificial-diffusivity approach of Cook…
This paper deals with the systematic development of structure-preserving approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic…
We study dynamics of the shearless stratified turbulent flows. Using the method of differential constraints we find a class of explicit solutions to the problem under consideration and establish that the differential constraint obtained…
We investigate the behaviour of local perturbations of a wide class of geometric PDEs on holomorphic Hermitian vector bundles over a compact complex manifold. Our main goal is to study the existence of solutions near an initial solution…
The simulation of fluid flow problems, specifically incompressible flows governed by the Navier-Stokes equations (NSE), holds fundamental significance in a range of scientific and engineering applications. Traditional numerical methods…
In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under…
We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a…
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…
The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by…
Occupied diffusions offer a Markovian framework for path-dependent dynamics by lifting the state space with a flow of occupation measures. Because this additional feature is infinite-dimensional, the simulation of these processes remains…
This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of…
We address several concerns related to the derivation of drift-ordered fluid equations. Starting from a fully Galilean invariant fluid system, we show how consistent sets of perturbative drift-fluid equations in the case of a isothermal…
The propagation of internal gravity waves in stratified media, such as those found in ocean basins and lakes, leads to the development of geometrical patterns called "attractors". These structures accumulate much of the wave energy and make…