scholarly journals PARAMETER-UNIFORM IMPROVED HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PROBLEMS WITH INTERIOR LAYERS

2018 ◽  
Vol 23 (2) ◽  
pp. 167-189 ◽  
Author(s):  
Kaushik Mukherjee
Keyword(s):  
Numerical Scheme ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Supremum Norm ◽  
Hybrid Scheme ◽  
Singularly Perturbed Problems ◽  
Central Difference ◽  
Convection Diffusion ◽  
Interior Layers ◽  
Wave Phenomena

In this paper, we consider a class of singularly perturbed convection-diffusion boundary-value problems with discontinuous convection coefficient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to difficulty while solving such problems using the classical numerical methods (standard central difference or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a significant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence.

A Parameter Robust Finite Element Method for Fourth Order Singularly Perturbed Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 337-349 ◽  
Author(s):  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Hermite Polynomials ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Singularly Perturbed Problems ◽  
Exponentially Graded ◽  
Element Method

AbstractWe consider fourth order singularly perturbed problems in one-dimension and the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$ defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error is measured in both the energy norm and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings through numerical computations, including a comparison with another scheme from the literature.

THE PARAMETER-ROBUST NUMERICAL METHOD BASED ON DEFECT-CORRECTION TECHNIQUE FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR

2010 ◽  
Vol 07 (04) ◽  
pp. 573-594 ◽  
Author(s):  
JUGAL MOHAPATRA ◽  
SRINIVASAN NATESAN
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Perturbation Parameter ◽  
Correction Method ◽  
Singularly Perturbed ◽  
Defect Correction ◽  
Central Difference ◽  
Correction Technique ◽  
Delay Differential

In this article, we consider a defect-correction method based on finite difference scheme for solving a singularly perturbed delay differential equation. We solve the equation using upwind finite difference scheme on piecewise-uniform Shishkin mesh, then apply the defect-correction technique that combines the stability of the upwind scheme and the higher-order central difference scheme. The method is shown to be convergent uniformly in the perturbation parameter and almost second-order convergence measured in the discrete maximum norm is obtained. Numerical results are presented, which are in agreement with the theoretical findings.

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.

Numerical analysis and simulation of delay parabolic partial differential equation involving a small parameter

2019 ◽  
Vol 37 (1) ◽  
pp. 289-312 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Content Type ◽  
Convection Diffusion ◽  
Initial Boundary Value

Purpose The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem. Design/methodology/approach The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh. Findings The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature. Originality/value A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.

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