On the convergence rate of nonstationary Navier-Stokes approximations

1980 ◽  
pp. 425-449 ◽  
Author(s):  
Reimund Rautmann
Keyword(s):  
Convergence Rate ◽  
Navier Stokes

scholarly journals Unsteady 2D and 3D Navier-Stokes Solver with Application of Multigrid Scheme to Pressure Poisson Fractional Step on Arbitrary Unstructured Grids in Various Applications with Emphasis on Ship Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-28
Author(s):  
Mehdi Pourmostafa ◽  
Parviz Ghadimi
Keyword(s):  
Convergence Rate ◽  
Poisson Equation ◽  
Multigrid Method ◽  
Stokes Equations ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Finite Volume Scheme ◽  
Fractional Step ◽  
Fractional Step Method ◽  
Pressure Poisson Equation

A 3D unsteady computer solver is presented to compute incompressible Navier-Stokes equations combined with the volume of fraction (VOF) method on an arbitrary unstructured domain. This is done to simulate fluid flows in various applications, especially around a marine vessel. The Navier-Stokes solver is based on the fractional steps method coupled with a finite volume scheme and collocated grids by which velocity components and pressure fields are defined at the center of the control volume. However, the fluxes are defined at the midpoint on their corresponding cell faces. On the other hand, the CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes) scheme is applied to capture the free surface. In the presented fractional step method, the pressure Poisson equation suffers from poor convergence rate by simple iterative methods like Successive Overrelaxation (SOR), especially in simulating complex geometrics like a ship with appendages. Therefore, to accelerate the convergence rate, an agglomeration multigrid method is applied on arbitrary moving mesh for solving pressure Poisson equation with two well-known cycles, V and W. In order to maintain accuracy, the geometry details should not change in grid coarsening procedure. Therefore, the boundary faces are assumed to be fixed in all grids level. This assumption requires nonstandard cells in coarsening procedures. To investigate the performance of the applied algorithm, various flows including one and two-phase flows are studied in two and three dimensions. It is found that the multigrid method can speed up the convergence rate of fractional step twofold. In most cases (not all), W cycle displays better performance. It is also concluded that the efficiency of the cycle depends on the number of meshes and complexity of the problem and this is mainly due to the data transferring between grids. Therefore, the type of cycle should be selected judiciously and carefully, while considering the mesh size and flow properties.

scholarly journals CONVERGENCE RATE FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH EXTERNAL FORCE

2006 ◽  
Vol 03 (03) ◽  
pp. 561-574 ◽  
Author(s):  
SEIJI UKAI ◽  
TONG YANG ◽  
HUIJIANG ZHAO
Keyword(s):  
Convergence Rate ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Optimal Convergence Rate ◽  
Potential Force ◽  
Stationary Potential ◽  
Navier Stokes Equations ◽  
The Stability ◽  
External Forces

For the compressible Navier–Stokes equations with a stationary potential force, the stability of the stationary solutions was studied by Matsumura and Nishida. The convergence rate to the stationary solutions in time was later studied by Deckelnick which was improved by Shibata and Tanaka for more general external forces. This paper deals with the case for the stationary potential force under some smallness condition, to establish an almost optimal convergence rate in L2(ℝN)-norm for N ≥ 3.

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