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The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations --…

Numerical Analysis · Mathematics 2024-12-20 Robert Altmann , Jan Heiland

In this work we propose a novel Hybrid High-Order method for the incompressible Navier--Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous…

Numerical Analysis · Mathematics 2018-10-11 Lorenzo Botti , Daniele Di Pietro , Jérôme Droniou

We develop and analyze a highly efficient, second-order time-marching scheme for infinite-dimensional nonlinear geophysical fluid models, designed to accurately approximate invariant measures-that is, the stationary statistical properties…

Numerical Analysis · Mathematics 2025-10-08 Daozhi Han , Xiaoming Wang

We propose a Hybrid High-Order (HHO) formulation of the incompressible Navier-Stokes equations with variable density that provides exact conservation of volume and, accordingly, pure advection of the density variable. The spatial…

Fluid Dynamics · Physics 2026-05-14 Lorenzo Botti , Francesco Carlo Massa

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order…

Numerical Analysis · Mathematics 2017-11-27 Marian Piatkowski , Steffen Müthing , Peter Bastian

A new type of low-regularity integrator is proposed for Navier-Stokes equations, coupled with a stabilized finite element method in space. Unlike the other low-regularity integrators for nonlinear dispersive equations, which are all fully…

Numerical Analysis · Mathematics 2021-07-29 Buyang Li , Shu Ma , Katharina Schratz

We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure…

Numerical Analysis · Mathematics 2019-11-07 John A. Evans , David Kamensky , Yuri Bazilevs

This paper presents the variational discretization of the compressible Navier-Stokes-Fourier system, in which the viscosity and the heat conduction terms are handled within the variational approach to nonequilibrium thermodynamics as…

Differential Geometry · Mathematics 2022-02-09 Benjamin Couéraud , François Gay-Balmaz

In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency…

Numerical Analysis · Mathematics 2016-06-29 Christoph Lehrenfeld , Joachim Schöberl

In this paper we propose a novel arbitrary high order accurate semi-implicit space-time DG method for the solution of the three-dimensional incompressible Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As…

Numerical Analysis · Mathematics 2016-06-22 Maurizio Tavelli , Michael Dumbser

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two…

Numerical Analysis · Mathematics 2019-08-02 Carmen Gräßle , Michael Hinze , Jens Lang , Sebastian Ullmann

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak monotonicity property, which implies that…

Numerical Analysis · Mathematics 2023-10-26 Hao Li , Shusen Xie , Xiangxiong Zhang

We generalize the explicit high-order positivity-preserving entropy stable spectral collocation schemes developed in Upperman 2023 and Yamaleev 2023 for the three-dimensional (3D) compressible Navier Stokes equations to a time implicit…

Numerical Analysis · Mathematics 2026-03-25 Mohammed Sayyari , Nail K. Yamaleev

In this work we introduce and analyze a novel Hybrid High-Order method for the steady incompressible Navier-Stokes equations. The proposed method is inf-sup stable on general polyhedral meshes, supports arbitrary approximation orders, and…

Numerical Analysis · Mathematics 2018-02-26 Daniele A. Di Pietro , Stella Krell

In this paper, we are interested in constructing a scheme solving compressible Navier--Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy…

Numerical Analysis · Mathematics 2023-09-13 Chen Liu , Xiangxiong Zhang

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order…

Computational Physics · Physics 2018-01-11 Oscar Bruno , Max Cubillos

The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier-Stokes equations coupled with a moving solid. This method presents the advantage to predict an…

Numerical Analysis · Mathematics 2017-07-12 Sébastien Court , Michel Fournié

A global time-discretized scheme for the Navier-Stokes equation system in its Leray projection form is defined. It is shown that the scheme converges to a bounded global classical solution for smooth data which have polynomial decay at…

Analysis of PDEs · Mathematics 2012-07-12 Joerg Kampen

High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with…

Numerical Analysis · Mathematics 2024-01-30 Charles J. Naudet , Matthew J. Zahr

In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are studied. Schemes proposed previously can at most achieve temporal accuracy of order…

Numerical Analysis · Mathematics 2015-06-17 Zhibo Wang , Seakweng Vong