scholarly journals Grid approximation of a singularly perturbed boundary value problem modelling heat transfer in the case of flow over a flat plate with suction of the boundary layer

2004 ◽  
Vol 166 (1) ◽  
pp. 221-232 ◽  
Author(s):  
J.J.H. Miller ◽  
G.I. Shishkin ◽  
B. Koren ◽  
L.P. Shishkina
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Flat Plate ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Grid Approximation

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

scholarly journals NUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM INCLUDING BOTH DELAY AND BOUNDARY LAYER

2018 ◽  
Vol 23 (4) ◽  
pp. 568-581 ◽  
Author(s):  
Erkan Cimen
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Delay Differential Equation ◽  
Order Parameter ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
First Order ◽  
Delay Differential

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.

GRID APPROXIMATION OF SINGULARLY PERTURBED PARABOLIC EQUATIONS WITH MOVING BOUNDARY LAYERS

2008 ◽  
Vol 13 (3) ◽  
pp. 421-442
Author(s):  
Grigorii Shishkin
Keyword(s):  
Boundary Value Problem ◽  
Moving Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Positive Direction ◽  
Grid Approximation ◽  
Necessary And Sufficient

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary layer appears in a neighbourhood of the left lateral boundary SL 1. It turns out that, in the class of difference schemes on rectangular grids condensing in a neighbourhood of SL 1 with respect to x and t, there do not exist schemes that converge even under the condition P 0 −1 Â ϵ1/2, where P 0 is the total number of nodes in the meshes used, that is, P 0 Â N N 0, where the values N and N 0 define the numbers of mesh points in x and t. On such meshes, convergence under the condition N −1 + N 0 −1 ≤ ϵ1/4 cannot be achieved. Examination of widths similar to Kolmogorov's widths allows us to establish necessary and sufficient conditions for the ϵ‐uniform convergence of approximations to the solution of the boundary value problem. Using these conditions, a scheme is constructed on a mesh being piece‐wise uniform in a coordinate system adapted to the moving boundary. This scheme converges ϵ‐uniformly at the rate O(N −1 ln N + N0 −1).

Experimental study of heat transfer in the boundary layer of a flat plate with the impact of weak shock waves on the leading edge

2020 ◽  
Author(s):  
V. L. Kocharin ◽  
A. A. Yatskikh ◽  
D. S. Prishchepova ◽  
A. V. Panina ◽  
Yu. G. Yermolaev ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Experimental Study ◽  
Shock Waves ◽  
Flat Plate ◽  
Leading Edge ◽  
Weak Shock ◽  
The Impact
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