Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments

2016 ◽  
Vol 75 (1) ◽  
pp. 113-145 ◽  
Author(s):  
Komal Bansal ◽  
Kapil K. Sharma
Keyword(s):  
Numerical Scheme ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Diffusion Reaction ◽  
Convection Diffusion

scholarly journals Fitted numerical scheme for singularly perturbed convection-diffusion reaction problems involving delays

2021 ◽  
pp. 6-6
Author(s):  
Mesfin Woldaregay ◽  
Worku Aniley ◽  
Gemechis Duressa
Keyword(s):  
Numerical Scheme ◽  
Numerical Test ◽  
Singularly Perturbed ◽  
Diffusion Reaction ◽  
Convection Diffusion ◽  
Solution Methods ◽  
Solution Bound ◽  
The Stability ◽  
Delay Differential ◽  
Exponential Boundary Layer

This paper deals with solution methods for singularly perturbed delay differential equations having delay on the convection and reaction terms. The considered problem exhibits an exponential boundary layer on the left or right side of the domain. The terms with the delay are treated using Taylor?s series approximation and the resulting singularly perturbed boundary value problem is solved using a specially designed exponentially finite difference method. The stability of the scheme is analysed and investigated using a comparison principle and solution bound. The formulated scheme converges uniformly with linear order of convergence. The theoretical findings are validated using three numerical test examples.

scholarly journals An adaptive mesh method for time dependent singularly perturbed differential-difference equations

2019 ◽  
Vol 8 (1) ◽  
pp. 328-339
Author(s):  
P. Pramod Chakravarthy ◽  
Kamalesh Kumar
Keyword(s):  
Prior Information ◽  
Grid Method ◽  
Adaptive Mesh ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Mesh Method

Abstract In this paper, a time dependent singularly perturbed differential-difference convection-diffusion equation is solved numerically by using an adaptive grid method. Similar boundary value problems arise in computational neuroscience in determination of the behaviour of a neuron to random synaptic inputs. The mesh is constructed adaptively by using the concept of entorpy function. In the proposed scheme, prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation free solution, without any user introduced parameters. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems

2016 ◽  
Vol 20 (5) ◽  
pp. 1340-1358 ◽  
Author(s):  
Yanping Chen ◽  
Li-Bin Liu
Keyword(s):  
Numerical Solution ◽  
Time Derivative ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Uniform Mesh ◽  
Convection Diffusion ◽  
Monitor Function ◽  
Backward Euler ◽  
Diffusion Problems

AbstractIn this paper, we study the numerical solution of singularly perturbed time-dependent convection-diffusion problems. To solve these problems, the backward Euler method is first applied to discretize the time derivative on a uniform mesh, and the classical upwind finite difference scheme is used to approximate the spatial derivative on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all temporal levels, we construct a positive monitor function, which is similar to the arc-length monitor function. Furthermore, the ε-uniform convergence of the fully discrete scheme is derived for the numerical solution. Finally, some numerical results are given to support our theoretical results.

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