scholarly journals An Efficient Computational Method for Singularly Perturbed Delay Parabolic Partial Differential Equations

2021 ◽  
Vol 15 ◽  
pp. 105-117
Author(s):  
Imiru Takele Daba ◽  
Gemechis File Duressa
Keyword(s):  
Numerical Scheme ◽  
Euler Method ◽  
Computational Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Fine Mesh ◽  
Theoretical Predictions ◽  
Uniformly Convergent

In this communication, a parameter uniform numerical scheme is proposed to solve singularly perturbed delay parabolic convection-diffusion equations. Taylor’s series expansion is applied to approximate the shift term. Then the resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for temporal discretization on uniform mesh and hybrid numerical scheme based on a midpoint upwind scheme in the coarse mesh regions and a cubic spline method in the fine mesh regions on a piecewise uniform Shishkin mesh for the spatial discretization. The proposed numerical scheme is shown to be an ε−uniformly convergent accuracy of first-order in time and almost second-order in space directions. Some test examples are considered to testify the theoretical predictions.

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.

scholarly journals A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Fasika Wondimu Gelu ◽  
Gemechis File Duressa
Keyword(s):  
Collocation Method ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Euler Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
Step Size ◽  
Uniformly Convergent ◽  
Type Boundary

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε -uniformly convergent of O Δ t + N − 2 ln 2 N , where Δ t and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ . Some numerical results are carried out to support the theory.

scholarly journals A Second Order Weighted Numerical Scheme on Nonuniform Meshes for Convection Diffusion Parabolic Problems

2020 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Time Derivative ◽  
Perturbation Parameter ◽  
Rectangular Domain ◽  
Shishkin Mesh ◽  
Second Order ◽  
Parabolic Problems ◽  
Uniform Mesh ◽  
Central Difference ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

In this article, a singularly perturbed parabolic convection-diffusion equation on a rectangular domain is considered. The solution of the problem possesses regular boundary layer which appears in the spatial variable. To discretize the time derivative, we use two type of schemes, first the implicit Euler scheme and second the implicit trapezoidal scheme on a uniform mesh. For approximating the spatial derivatives, we use the monotone hybrid scheme, which is a combination of midpoint upwind scheme and central difference scheme with variable weights on Shishkin-type meshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh and modified Bakhvalov-Shishkin mesh). We prove that both numerical schemes converge uniformly with respect to the perturbation parameter and are of second order accurate. Thomas algorithm is used to solve the tri-diagonal system. Finally, to support the theoretical results, we present a numerical experiment by using the proposed methods.

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