CONVERGENCE RATES IN H2,r OF ROTHE'S METHOD TO THE NAVIER-STOKES EQUATIONS

1998 ◽  
pp. 127-135
Author(s):  
R. Rautmann
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Rothe's Method ◽  
Rothe’S Method

Simultaneous Variable Solutions of the Incompressible Steady Navier-Stokes Equations in General Curvilinear Coordinate Systems

1992 ◽  
Vol 114 (3) ◽  
pp. 299-305 ◽  
Author(s):  
G. Vradis ◽  
V. Zalak ◽  
J. Bentson
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Flow Direction ◽  
Solution Algorithm ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Solution Technique ◽  
Algebraic Equations ◽  
Coordinate Systems ◽  
Navier Stokes Equations

A simultaneous variable solution technique for the incompressible, steady, two-dimensional Navier-Stokes equations in primitive formulation and general curvilinear orthogonal and nonorthogonal coordinate systems has been developed. The governing equations are discretized using finite difference approximations. The formulation is fully second order accurate and the well-known staggered grid of Welch and Harlow is used. The solution algorithm is based on an iterative marching technique in which the algebraic equations are linearized by evaluating the coefficients at the previous iteration level. The resulting system of linear equations is solved in a marching fashion by employing a block tridiagonal solution algorithm to obtain the solution along lines transverse to the main flow direction. The strong pressure-velocity coupling inherent in the present formulation results in high convergence rates. Flows in channels of different geometries have been computed and the results have been compared to available data in the literature. In all cases the method has demonstrated to be accurate, robust and computationally efficient.

A HYBRID FINITE-ANALYTIC ALGORITHM FOR SOLVING THREE-DIMENSIONAL ADVECTION-DIFFUSION EQUATIONS

2004 ◽  
Vol 01 (03) ◽  
pp. 407-430 ◽  
Author(s):  
H. M. HU ◽  
K.-H. WANG
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Diffusion Equations ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
The Stability

The hybrid finite-analytic (HFA) method for discretization of a three-dimensional advection-diffusion equation is developed using the superposition of the HFA solutions of locally linearized one-dimensional advection-diffusion equations. An example calculation of a system of three-dimensional nonlinear equations is conducted to test the convergence and accuracy of the 7-point numerical scheme. Good agreements between calculated and analytical solutions are obtained. An algorithm based on the HFA method with multigrid technique and Gauss-Seidel iteration is also developed to solve the three-dimensional Navier-Stokes equations in a staggered grid system. The stability and efficiency of the method are demonstrated by performing calculations of the fluid flow in a three-dimensional cubic cavity with a moving top wall. The proposed procedure is observed to exhibit good rates of smoothing and almost grid-independent convergence rates in comparison with a single-grid iteration method. The results are in excellent agreement with other published computational results.

scholarly journals Convergence rates for the numerical approximation of the 2D stochastic Navier–Stokes equations

2021 ◽  
Vol 147 (3) ◽  
pp. 553-578
Author(s):  
Dominic Breit ◽  
Alan Dodgson
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Linear Convergence ◽  
Careful Analysis ◽  
Two Dimensions ◽  
Convergence In Probability ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cylindrical Wiener Process ◽  
Numer Anal

AbstractWe study stochastic Navier–Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measured in the $$L^\infty _tL^2_x\cap L^2_tW^{1,2}_x$$ L t ∞ L x 2 ∩ L t 2 W x 1 , 2 -norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (SIAM J Numer Anal 50(5):2467–2496, 2012) where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.

OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH POTENTIAL FORCES

2007 ◽  
Vol 17 (05) ◽  
pp. 737-758 ◽  
Author(s):  
RENJUN DUAN ◽  
SEIJI UKAI ◽  
TONG YANG ◽  
HUIJIANG ZHAO
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Initial Perturbation ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Potential Force ◽  
Navier Stokes Equations ◽  
Optimal Convergence ◽  
Whole Space ◽  
Optimal Convergence Rates

For the viscous and heat-conductive fluids governed by the compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp - Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.

Converge rates towards stationary solutions for the outflow problem of planar magnetohydrodynamics on a half line

2019 ◽  
Vol 149 (5) ◽  
pp. 1291-1322 ◽  
Author(s):  
Haiyan Yin
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Weighted Sobolev Space ◽  
Initial Perturbation ◽  
Global Solutions ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weighted Energy Method ◽  
The Relationship

AbstractIn this paper, convergence rates of solutions towards stationary solutions for the outflow problem of planar magnetohydrodynamics (MHD) are investigated. Inspired by the relationship between MHD and Navier-Stokes, we prove that the global solutions of the planar MHD converge to the corresponding stationary solutions of Navier-Stokes equations. We obtain the corresponding convergence rates based on the weighted energy method when the initial perturbation belongs to some weighted Sobolev space.

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