scholarly journals An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

2009 ◽  
Vol 233 (4) ◽  
pp. 1139-1154 ◽  
Author(s):  
Marco Picasso ◽  
Virabouth Prachittham
Keyword(s):  
Adaptive Algorithm ◽  
Diffusion Problem ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1878
Author(s):  
Eyaya Fekadie Anley ◽  
Zhoushun Zheng
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Diffusion Problem ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Finite Domain ◽  
Two Dimensional ◽  
Convection Diffusion ◽  
Nicolson Scheme ◽  
Crank Nicolson

In this paper, we have considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. The convection and diffusion equation can depend on both spatial and temporal variables. Crank-Nicolson scheme for time combined with weighted and shifted Grünwald-Letnikov difference operator for space are implemented to get second order convergence both in space and time. Unconditional stability and convergence order analysis of the scheme are explained theoretically and experimentally. The numerical tests are indicated that the Crank-Nicolson scheme with weighted shifted Grünwald-Letnikov approximations are effective numerical methods for two dimensional two-sided space fractional convection-diffusion equation.

scholarly journals A NOTE ON THE NUMERICAL APPROACH FOR THE REACTION–DIFFUSION PROBLEM WITH A FREE BOUNDARY CONDITION

2010 ◽  
Vol 51 (3) ◽  
pp. 317-330
Author(s):  
E. ÖZUĞURLU
Keyword(s):  
Differential Equation ◽  
Free Boundary ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Liquid Films ◽  
Diffusion Problem ◽  
Nicolson Scheme ◽  
Crank Nicolson

AbstractThe equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

scholarly journals A NEW DOMAIN DECOMPOSITION PARALLEL ALGORITHM FOR CONVECTION–DIFFUSION PROBLEM

2014 ◽  
Vol 19 (4) ◽  
pp. 589-605 ◽  
Author(s):  
Jiansong Zhang ◽  
Danping Yang
Keyword(s):  
Approximate Solution ◽  
Domain Decomposition ◽  
Parallel Algorithm ◽  
Mesh Size ◽  
Diffusion Problem ◽  
Time Dependent ◽  
Time Step ◽  
Convection Diffusion ◽  
Overlapping Domain Decomposition ◽  
Iteration Number

Basing on overlapping domain decomposition, we construct a new parallel algorithm combined the method of subspace correction with least-squares procedure for solving time-dependent convection–diffusion problem. This algorithm is fully parallel. We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to given accuracy at each time step.

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