Some linear stability results for iterative schemes for implicit Runge-Kutta methods

1996 ◽  
Vol 36 (1) ◽  
pp. 77-85 ◽  
Author(s):  
G. J. Cooper
Keyword(s):  
Linear Stability ◽  
Iterative Schemes ◽  
Runge Kutta ◽  
Stability Results

Comparison of linear stability results with flight transition data

AIAA Journal ◽  
1995 ◽  
Vol 33 (1) ◽  
pp. 161-163 ◽  
Author(s):  
J. A. Masad ◽  
M. R. Malik
Keyword(s):  
Linear Stability ◽  
Stability Results

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Sediment-laden fresh water above salt water: linear stability analysis

2011 ◽  
Vol 691 ◽  
pp. 279-314 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Fresh Water ◽  
Settling Velocity ◽  
Interface Thickness ◽  
Particle Settling ◽  
Salinity Field ◽  
Unstable Layer ◽  
Stability Results ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

Linear stability analysis for periodic travelling waves of the Boussinesq equation and the Klein–Gordon–Zakharov system

2014 ◽  
Vol 144 (3) ◽  
pp. 455-489 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Milena Stanislavova ◽  
Atanas Stefanov
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Wide Class ◽  
Boussinesq Equation ◽  
Travelling Waves ◽  
Periodic Waves ◽  
Zakharov System ◽  
Klein Gordon ◽  
Spatially Periodic ◽  
Stability Results

The question of the linear stability of spatially periodic waves for the Boussinesq equation (in the cases p = 2, 3) and the Klein–Gordon–Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability) when the perturbations are taken with the same period T. In particular, our results allow us to completely recover the linear stability results, in the limit T → ∞, for the whole-line case.

scholarly journals Convergence and Stability of New Approximation Algorithms for Certain Contractive-Type Mappings

2021 ◽  
Vol 2 ◽  
pp. 1
Author(s):  
Imo Kalu Agwu ◽  
Donatus Ikechi Igbokwe
Keyword(s):  
Approximation Algorithms ◽  
Fixed Points ◽  
Iterative Scheme ◽  
Type Condition ◽  
Iterative Schemes ◽  
Convergence And Stability ◽  
Contractive Type ◽  
Stability Results

We present new fixed points algorithms called multistep H-iterative scheme and multistep SH-iterative scheme. Under certain contractive-type condition, convergence and stability results were established without any imposition of the ’sum conditions’, which to a large extent make some existing iterative schemes so far studied by other authors in this direction practically inefficient. Our results complement and improve some recent results in literature.

scholarly journals Linear Stability of Partitioned Runge–Kutta Methods

2011 ◽  
Vol 49 (1) ◽  
pp. 232-263 ◽  
Author(s):  
R. I. McLachlan ◽  
Y. Sun ◽  
P. S. P. Tse
Keyword(s):  
Linear Stability ◽  
Runge Kutta

Linear stability of WENO schemes coupled with explicit Runge-Kutta schemes

2011 ◽  
Vol 69 (6) ◽  
pp. 1065-1095 ◽  
Author(s):  
V. Hermes ◽  
I. Klioutchnikov ◽  
H. Olivier
Keyword(s):  
Linear Stability ◽  
Weno Schemes ◽  
Runge Kutta
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