Incomplete Iterative Implicit Schemes

2020 ◽  
Vol 20 (4) ◽  
pp. 727-737 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Approximate Solution ◽  
Iterative Methods ◽  
Parabolic Equations ◽  
Evolutionary Equation ◽  
Finite Dimensional ◽  
Implicit Schemes ◽  
Multidimensional Problems ◽  
Computational Implementation ◽  
The Stability ◽  
Finite Dimensional Hilbert Space

AbstractIn numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

scholarly journals Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Dumitru Baleanu
Keyword(s):  
Parabolic Equation ◽  
Fractional Calculus ◽  
Numerical Solution ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Implicit Schemes ◽  
The Stability

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

Oscillations-Induced Transitions and Their Application in Control of Dynamical Systems

1990 ◽  
Vol 112 (3) ◽  
pp. 313-319 ◽  
Author(s):  
J. Bentsman
Keyword(s):  
Dynamical Systems ◽  
Parabolic Equations ◽  
Closed Loop ◽  
Distributed Parameter ◽  
Control Laws ◽  
Loop Control ◽  
Finite Dimensional ◽  
Analytical Tools ◽  
Semilinear Parabolic ◽  
Qualitative Changes

Studies of the use of oscillations for control purposes continue to reveal new practically important properties unique to the oscillatory open and closed loop control laws. The goal of this paper is to enlarge the available set of analytical tools for such studies by introducing a method of analysis of the qualitative changes in the behavior of dynamical systems caused by the zero mean parametric excitations. After summarizing and slightly refining a technique developed previously for the finite dimensional nonlinear systems, we consider an extension of this technique to a class of distributed parameter systems (DPS) governed by semilinear parabolic equations. The technique presented is illustrated by several examples.

On the stability of viscous flow in a curved channel

1958 ◽  
Vol 244 (1237) ◽  
pp. 186-198 ◽  
Keyword(s):  
Approximate Solution ◽  
Pressure Gradient ◽  
Viscous Flow ◽  
Close Agreement ◽  
Curved Channel ◽  
A Value ◽  
Concentric Cylinders ◽  
The Stability ◽  
Pattern Of Motion

In this paper the stability of viscous flow between two concentric cylinders due to a pressure gradient acting round the cylinders is considered when the spacing between the cylinders is small compared with their radii. Two methods of approximate solution are described, both of which show that instability first sets in when the parameter R √( d / R 1 ) attains a value of about 36 in close agreement with earlier results of Dean (1928). The pattern of motion which then sets in is of the familar cellular type but with a marked asymmetry.

FRACTIONAL ORDER ANALYSIS OF HBV AND HCV CO-INFECTION UNDER ABC DERIVATIVE

Fractals ◽  
2021 ◽  
Author(s):  
HUSSAM ALRABAIAH ◽  
MATI UR RAHMAN ◽  
IBRAHIM MAHARIQ ◽  
SAMIA BUSHNAQ ◽  
MUHAMMAD ARFAN
Keyword(s):  
Mathematical Model ◽  
Fixed Point ◽  
Approximate Solution ◽  
Fractional Order ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Comparative Result ◽  
Order Analysis ◽  
The Stability ◽  
Fractional Orders

In this paper, we consider a fractional mathematical model describing the co-infection of HBV and HCV under the non-singular Mittag-Leffler derivative. We also investigate the qualitative analysis for at least one solution and a unique solution by applying the approach fixed point theory. For an approximate solution, the technique of the iterative fractional order Adams–Bashforth scheme has been implemented. The simulation for the proposed scheme has been drawn at various fractional order values lying between (0,1) and integer-order of 1 via using Matlab. All the compartments have shown convergence and stability with time. A detailed comparative result has been given by the different fractional orders, which showed that the stability was achieved more rapidly at low orders.

scholarly journals Stability of Leapfrog Constant-Coefficients Semi-Implicit Schemes for the Fully Elastic System of Euler Equations: Case with Orography

2005 ◽  
Vol 133 (5) ◽  
pp. 1065-1075 ◽  
Author(s):  
P. Bénard ◽  
J. Mašek ◽  
P. Smolíková
Keyword(s):  
Euler Equations ◽  
Evolution Equations ◽  
Momentum Equation ◽  
Primitive Equations ◽  
Prognostic Variables ◽  
Flat Terrain ◽  
Implicit Schemes ◽  
Constant Coefficients ◽  
The Stability ◽  
The One

Abstract The stability of constant-coefficients semi-implicit schemes for the hydrostatic primitive equations and the fully elastic Euler equations in the presence of explicitly treated thermal residuals has been theoretically examined in the earlier literature, but only for the case of a flat terrain. This paper extends these analyses to a case in which an orography is present, in the shape of a uniform slope. It is shown, with mass-based coordinates, that for the Euler equations, the presence of a slope reduces furthermore the set of the prognostic variables that can be used in the vertical momentum equation to maintain the robustness of the scheme, compared to the case of a flat terrain. The situation appears to be similar for systems cast in mass-based and height-based vertical coordinates. Still for mass-based vertical coordinates, an optimal prognostic variable is proposed and is shown to result in a robustness similar to the one observed for the hydrostatic primitive equations system. The prognostic variables that lead to robust semi-implicit schemes share the property of having cumbersome evolution equations, and an alternative time treatment of some terms is then required to keep the evolution equation reasonably simple. This treatment is shown not to modify substantially the stability of the time scheme. This study finally indicates that with a pertinent choice for the prognostic variables, mass-based vertical coordinates are equally suitable as height-based coordinates for efficiently solving the compressible Euler equations system.

scholarly journals Stability of a Hot Smoluchowski Fluid

2001 ◽  
Vol 08 (01) ◽  
pp. 19-27 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Numerical Range ◽  
Nonlinear Parabolic Equations ◽  
Stationary Solutions ◽  
Small Perturbations ◽  
Material Density ◽  
Nonlinear Parabolic ◽  
The Stability ◽  
Special Case

We study coupled nonlinear parabolic equations for a fluid described by a material density ρ and a temperature Θ, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearized equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for Θ and no-flow conditions for the material. The spectrum of the generator L of time evolution, regarded as an operator on L2[0,1], is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is ℂ.

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