A Uniformly Convergent Numerical Algorithm on Harmonic ($$H(\ell )$$) Mesh for Parabolic Singularly Perturbed Convection-Diffusion Problems with Boundary Layer

2022 ◽  
Author(s):  
Gajendra Babu ◽  
M. Prithvi ◽  
Kapil K. Sharma ◽  
V. P. Ramesh
Keyword(s):  
Boundary Layer ◽  
Numerical Algorithm ◽  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Uniformly Convergent ◽  
Diffusion Problems

scholarly journals Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Gemechis File Duressa
Keyword(s):  
Boundary Layer ◽  
Hybrid Method ◽  
Singularly Perturbed ◽  
Nonoscillatory Solutions ◽  
Convection Diffusion ◽  
Layer Region ◽  
The Stability ◽  
Regular Region ◽  
The Right ◽  
Diffusion Problems

This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost second-order rate of convergence.

scholarly journals ROBUST NOVEL HIGH-ORDER ACCURATE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION‐DIFFUSION PROBLEMS

2005 ◽  
Vol 10 (4) ◽  
pp. 393-412 ◽  
Author(s):  
G. I. Shishkin
Keyword(s):  
Numerical Methods ◽  
Order Of Convergence ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Order Of Accuracy ◽  
Convection Diffusion ◽  
Expansion Technique ◽  
Uniformly Convergent ◽  
Problem Data ◽  
Diffusion Problems

For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy. Straipsnyje nagrinejami nedidelio tikslumo ϵ‐tolygiai konvertuojantys skaitmeniniai metodai, singuliariai sutrikdytiems kraštiniams uždaviniams. Paraboliniam konvekcijos‐difuzijos uždaviniui konvergavimo eile neviršija vienos antrosios, jeigu uždavinio duomenys yra pakankamai glodūs. Tačiau trūkiems pradiniams duomenims eile yra ne aukštesne už 2−1. Šio tipo uždaviniams, naudojant naujai išvestus metodus, darbe sukonstruotos ϵ‐tolygiai konvertuojančios schemos aukštesniu tikslumu.

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