Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation

2012 ◽  
pp. 293-301 ◽  
Author(s):  
G. Shishkin
Keyword(s):  
Diffusion Equation ◽  
Difference Schemes ◽  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Stability Of Difference Schemes ◽  
Uniform Grids

OPTIMAL DIFFERENCE SCHEMES ON PIECEWISE‐UNIFORM MESHES FOR A SINGULARLY PERTURBED PARABOLIC CONVECTION‐DIFFUSION EQUATION

2008 ◽  
Vol 13 (1) ◽  
pp. 99-112 ◽  
Author(s):  
G. Shishkin
Keyword(s):  
Diffusion Equation ◽  
Transition Point ◽  
Difference Schemes ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Step Size ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Grid Approximation ◽  
Discrete Norm

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic convection‐diffusion equation. For this problem, upwind difference schemes on the well‐known piecewise‐uniform meshes converge ϵ‐uniformly in the maximum discrete norm at the rate O(N− 1lnN + N0 −1 ), where N + 1 and N 0 + 1 are the number of mesh points in x and t respectively; the number of nodes in the x‐mesh before the transition point (the point where the step‐size changes) and after it are the same. Under the condition N Â N 0 this scheme converges at the rate O(P−1/2 ln P); here P = (N + 1)(N 0 + 1) is the total number of nodes in the piecewise‐uniform mesh. Schemes on piecewise‐uniform meshes are constructed that are optimal with respect to the convergence rate. These schemes converge ϵ‐uniformly at the rate O(P−1/2 ln1/2 P). In optimal meshes based on widths that are similar to Kolmogorov's widths, the ratio of mesh points in x and t is of O((ϵ + ln−1 P)−1). Under the condition ϵ = o( 1), most nodes in such a mesh in x are placed before the transition point.

Spectral analysis of finite difference schemes for convection diffusion equation

2017 ◽  
Vol 150 ◽  
pp. 95-114 ◽  
Author(s):  
V.K. Suman ◽  
Tapan K. Sengupta ◽  
C. Jyothi Durga Prasad ◽  
K. Surya Mohan ◽  
Deepanshu Sanwalia
Keyword(s):  
Spectral Analysis ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Convection Diffusion ◽  
Convection Diffusion Equation
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