scholarly journals Analytical Construction of Uniformly Convergent Method for Convection Diffusion Problem

2021 ◽  
Vol 9 (3) ◽  
Author(s):  
Ali Filiz
Keyword(s):  
Adjoint Operator ◽  
Diffusion Problem ◽  
Convection Diffusion ◽  
Formal Adjoint ◽  
Analytical Construction ◽  
Convergent Method ◽  
Uniformly Convergent

In this paper, we study the uniformly convergent method on equidistant meshes for the convection-diffusion problem of type; where   the formal adjoint operator of L. Lu=-εu''+bu'+c u=f(x), u(0)=0, u(1)=0 At the end of the this paper we will generate the scheme; -e^(ρ_i )/(e^(ρ_i )+1) U_(i-1)+U_i-1/(e^(ρ_i )+1) U_(i+1)=(f_i-c_i U_i ) h/b ((e^(ρ_i )-1)/(e^(ρ_i )+1))

Tailored Finite Point Method for Parabolic Problems

2016 ◽  
Vol 16 (4) ◽  
pp. 543-562 ◽  
Author(s):  
Zhongyi Huang ◽  
Yi Yang
Keyword(s):  
Diffusion Problem ◽  
Point Method ◽  
Parabolic Problems ◽  
Finite Difference Schemes ◽  
Finite Point ◽  
Convection Diffusion ◽  
Finite Point Method ◽  
Tailored Finite Point Method ◽  
Uniformly Convergent ◽  
Tailored Finite Point

AbstractIn this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.

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.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

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