Numerical solution of parabolic problems by the generalized Crank-Nicolson scheme

CALCOLO ◽  
1975 ◽  
Vol 12 (1) ◽  
pp. 51-61 ◽  
Author(s):  
Nabil R. Nassif
Keyword(s):  
Numerical Solution ◽  
Parabolic Problems ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals Numerical solution of advection-diffusion equation using finite difference schemes

2020 ◽  
Vol 55 (1) ◽  
pp. 15-22
Author(s):  
LS Andallah ◽  
MR Khatun
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Finite Difference Schemes ◽  
Qualitative Behavior ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nicolson Scheme ◽  
And Diffusion ◽  
Crank Nicolson

This paper presents numerical simulation of one-dimensional advection-diffusion equation. We study the analytical solution of advection diffusion equation as an initial value problem in infinite space and realize the qualitative behavior of the solution in terms of advection and diffusion co-efficient. We obtain the numerical solution of this equation by using explicit centered difference scheme and Crank-Nicolson scheme for prescribed initial and boundary data. We implement the numerical scheme by developing a computer programming code and present the stability analysis of Crank-Nicolson scheme for ADE. For the validity test, we perform error estimation of the numerical scheme and presented the numerical features of rate of convergence graphically. The qualitative behavior of the ADE for different choice of the advection and diffusion co-efficient is verified. Finally, we estimate the pollutant in a river at different times and different points by using these numerical scheme. Bangladesh J. Sci. Ind. Res.55(1), 15-22, 2020

scholarly journals Solving Advection Equations by Applying the Crank-Nicolson Scheme Combined with the Richardson Extrapolation

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Zahari Zlatev ◽  
Ivan Dimov ◽  
István Faragó ◽  
Krassimir Georgiev ◽  
Ágnes Havasi ◽  
...  
Keyword(s):  
Mathematical Models ◽  
Numerical Solution ◽  
Large Scale ◽  
Richardson Extrapolation ◽  
Numerical Treatment ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Nicolson Scheme ◽  
Crank Nicolson

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations will be formulated and proved in this paper. The usefulness of the combination consisting of the Crank-Nicolson scheme and the Richardson Extrapolation will be illustrated by numerical examples.

Beam Propagation Method Based on the Iterated Crank–Nicolson Scheme for Solving Large-Scale Wave Propagation Problems

2015 ◽  
Vol 51 (3) ◽  
pp. 1-4 ◽  
Author(s):  
Dimitra A. Ketzaki ◽  
Ioannis T. Rekanos ◽  
Theodoros I. Kosmanis ◽  
Traianos V. Yioultsis
Keyword(s):  
Wave Propagation ◽  
Large Scale ◽  
Beam Propagation Method ◽  
Beam Propagation ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

2021 ◽  
Vol 5 (4) ◽  
pp. 274
Author(s):  
Jinfeng Wang ◽  
Baoli Yin ◽  
Yang Liu ◽  
Hong Li ◽  
Zhichao Fang
Keyword(s):  
Fourth Order ◽  
Mixed Finite Element ◽  
Calculated Data ◽  
Coupled System ◽  
Wave Model ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Wave ◽  
Nicolson Scheme ◽  
Crank Nicolson

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.

scholarly journals Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay

2021 ◽  
Vol 57 ◽  
pp. 128-141
Author(s):  
M. Ibrahim ◽  
V.G. Pimenov
Keyword(s):  
Piecewise Linear ◽  
Linear Interpolation ◽  
Fractional Diffusion ◽  
Dimensional System ◽  
Two Dimensional ◽  
Piecewise Linear Interpolation ◽  
Functional Delay ◽  
General Difference ◽  
Nicolson Scheme ◽  
Crank Nicolson

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

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