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The high-order numerical solution of the non-linear shallow water equations (and of hyperbolic systems in general) is susceptible to unphysical Gibbs oscillations that form in the proximity of strong gradients. The solution to this problem…

Numerical Analysis · Mathematics 2016-07-18 Simone Marras , Michal A. Kopera , Emil M. Constantinescu , Jenny Suckale , Francis X. Giraldo

This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, F. M. Huang, A. Matsumura and X. D. Shi showed that there exists viscous shock wave solution to the inflow…

Analysis of PDEs · Mathematics 2015-06-23 Dongfen Bian , Lili Fan , Lin He , Huijiang Zhao

We consider an initial value problem for shell models that mimic turbulent velocity fluctuations over a geometric sequence of scales. Our goal is to study the convergence of solutions in the inviscid (more generally, vanishing…

Analysis of PDEs · Mathematics 2025-08-07 Alexei A. Mailybaev

In this paper, we have solved 1D special relativistic hydrodynamical equations using different numerical method in computational gas dynamics. The numerical solutions of these equations for smooth wave cases give better solution when we use…

General Relativity and Quantum Cosmology · Physics 2007-05-23 Orhan Donmez

Nudging is an important data assimilation technique where partial field measurements are used to control the evolution of a dynamical system and/or to reconstruct the entire phase-space configuration of the supplied flow. Here, we apply it…

Fluid Dynamics · Physics 2020-02-12 P. Clark Di Leoni , A. Mazzino , L. Biferale

Conventional mathematical models for simulating incompressible fluid flow problems are based on the Navier-Stokes equations expressed in terms of pressure and velocity. In this context, pressure-velocity coupling is a key issue, and…

Mathematical Physics · Physics 2025-06-06 Ricardo Costa , Stéphane Clain , Gaspar J. Machado , João M. Nóbrega

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms which…

Numerical Analysis · Mathematics 2023-08-08 Michele Giuliano Carlino , Walter Boscheri

A Skeleton-stabilized ImmersoGeometric Analysis technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed formulation fits within the framework of the finite cell method, where essential…

We formulate and solve a generalized inverse Navier-Stokes problem for the joint velocity field reconstruction and boundary segmentation of noisy flow velocity images. To regularize the problem we use a Bayesian framework with Gaussian…

High-fidelity reconstruction of fluids from sparse multiview RGB videos remains a formidable challenge due to the complexity of the underlying physics as well as complex occlusion and lighting in captures. Existing solutions either assume…

In 1995, Kazhikhov and Vaigant introduced a particular class of isentropic compressible Navier-Stokes equations with variable viscosity coefficients and, for the first time, established the existence of global smooth solutions for…

Analysis of PDEs · Mathematics 2025-12-23 Jie Fan , Xiangdi Huang

We propose a fourth order Navier-Stokes solver based on the immersed interface method (IIM), for flow problems with stationary and one-way coupled moving boundaries and interfaces. Our algorithm employs a Runge-Kutta-based projection method…

Fluid Dynamics · Physics 2025-08-22 Xinjie Ji , Changxiao Nigel Shen , Wim M. van Rees

The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work…

Numerical Analysis · Mathematics 2025-06-04 Michael J. Facci , Ebrahim M. Kolahdouz , Boyce E. Griffith

The study of uncertainty propagation poses a great challenge to design numerical solvers with high fidelity. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction…

Computational Physics · Physics 2021-12-14 Tianbai Xiao , Jonas Kusch , Julian Koellermeier , Martin Frank

In the case of hyperbolic conservation laws, high-order methods, such as the classical DG method, experience the phenomenon of unwanted high-frequency oscillations in the vicinity of a shock. Shock-capturing methods such as artificial…

Numerical Analysis · Mathematics 2025-11-04 Sai Shruthi Srinivasan , Siva Nadarajah

We address a numerical instability that arises in the directionally split computation of hydrodynamic flows when shock fronts are parallel to a grid plane. Transverse oscillations in pressure, density and temperature are produced that are…

Astrophysics · Physics 2007-05-23 R. S. Sutherland , D. K. Bisset , G. V. Bicknell

In the recent paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power…

Analysis of PDEs · Mathematics 2019-11-21 Yongcai Geng , Yachun Li , Shengguo Zhu

It is shown here that a subset of the implicit analytical shock solutions discovered by Becker and by Johnson can be inverted, yielding several exact closed-form solutions of the one-dimensional compressible Navier-Stokes equations for an…

Fluid Dynamics · Physics 2014-03-28 Bryan M. Johnson

We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum $\rho_0 u_0$ belongs to the set of…

Analysis of PDEs · Mathematics 2019-01-11 Boris Haspot

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous…

Analysis of PDEs · Mathematics 2017-12-12 Marius Paicu , Ping Zhang