An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

1996 ◽  
Vol 12 (4) ◽  
pp. 407-421 ◽  
Author(s):  
F. Schieweck ◽  
L. Tobiska
Keyword(s):  
Error Estimate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Optimal Order ◽  
Order Error ◽  
Navier Stokes Equations ◽  
Upwind Discretization

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

scholarly journals Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

2020 ◽  
Author(s):  
David Maltese ◽  
Antonín Novotný
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Error Estimate ◽  
Strong Solution ◽  
Stokes Equations ◽  
Main Tool ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Mac Scheme

Abstract We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

CUDA Based Numerical Simulation of Cavity Flow and Performance Analysis

2013 ◽  
Vol 291-294 ◽  
pp. 1954-1957
Author(s):  
Xiao Ping Li ◽  
Hong Ming Zhang
Keyword(s):  
Stokes Equations ◽  
Cavity Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Floating Point Arithmetic ◽  
Upwind Discretization ◽  
C Program ◽  
And Performance ◽  
Volume Method ◽  
Point Arithmetic

Cuda has been widely used in computational fluid dynamics due to the powerful abilities of floating point arithmetic on gpu.This paper solved the Navier-Stokes equations of two dimensional incompressible flow using parallel programming on cuda. The finite volume method and the second-order upwind discretization scheme were used in the simulation.The speed of serial c program and the cuda based program were compared and we also compared the two programs on different hardware.The simulations got high precision results,which showed that the cuda based parallel computing is much more efficiency,and the parallel algorithm could get a more than 10 times the acceleration.

scholarly journals Reduced order methods for nonlinear parametric problems with branching solutions

2018 ◽  
Author(s):  
Federico Pichi ◽  
Martin Wilfried Hess ◽  
Annalisa Quaini ◽  
Gianluigi Rozza
Keyword(s):  
Stokes Equations ◽  
Spectral Element ◽  
Navier Stokes ◽  
Reduced Basis ◽  
State Equations ◽  
Order Error ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Plate Equations ◽  
Basis Method

The aim of this work is to show the applicability of the reduced basis model reduction in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e., following the different bifurcating branches, as well as determining the bifurcation point itself, is a complex computational task. Reduced Order Models (ROM) can potentially reduce the computational burden by several orders of magnitude, in particular in conjunction with sampling techniques. In the first task we focus on nonlinear structural mechanics, and we deal with an application of ROM to Von Kármán plate equations, where the buckling effect arises, adopting reduced basis method. Moreover, in the search of the bifurcation points, it is crucial to supplement the full problem with a reduced generalized parametric eigenvalue problem, properly paired with state equations and also a reduced order error analysis. These studies are carried out in view of vibroacoustic applications (in collaboration with A.T. Patera at MIT). As second task we consider the incompressible Navier-Stokes equations, discretized with the spectral element method, in a channel and a cavity. Both system undergo bifurcations with increasing Reynolds - and Grashof - number, respectively. Applications of this model are contraction-expansion channels, found in many biological systems, such as the human heart, for instance, or crystal growth in cavities, used in semiconductor production processes. This last task is in collaboration with A. Alla and M. Gunzburger (Florida State University).

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