Solving a variational parabolic equation with the periodic condition by a projection-difference method with the Crank–Nicolson scheme in time

2017 ◽  
Vol 58 (4) ◽  
pp. 591-599
Author(s):  
A. S. Bondarev ◽  
V. V. Smagin
Keyword(s):  
Parabolic Equation ◽  
Difference Method ◽  
Periodic Condition ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals A New Alternating Segment Crank-Nicolson Scheme for the Fourth-Order Parabolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ge-yang Guo ◽  
Bo Liu
Keyword(s):  
Parabolic Equation ◽  
Fourth Order ◽  
Unconditional Stability ◽  
Truncation Errors ◽  
Order Parabolic Equation ◽  
Intrinsic Parallelism ◽  
Fourth Order Parabolic Equation ◽  
The Stability ◽  
Nicolson Scheme ◽  
Crank Nicolson

A group of asymmetric difference schemes to approach the fourth-order parabolic equation is given. According to these schemes and the Crank-Nicolson scheme, an alternating segment Crank-Nicolson scheme with intrinsic parallelism is constructed. The truncation errors and the stability are discussed. Numerical simulations show that this new scheme has unconditional stability and high accuracy and convergency, and it is in preference to the implicit scheme method.

THE STRONG-NORM CONVERGENCE OF A PROJECTION-DIFFERENCE METHOD OF SOLUTION OF A PARABOLIC EQUATION WITH THE PERIODIC CONDITION ON THE SOLUTION

2018 ◽  
pp. 617-623
Author(s):  
Andrei Sergeevich Bondarev
Keyword(s):  
Exact Solution ◽  
Parabolic Equation ◽  
Difference Method ◽  
Approximate Solutions ◽  
Linear Parabolic Equation ◽  
Periodic Condition ◽  
Method Of Solution ◽  
The Galerkin Method ◽  
Convergence Of Approximate Solutions ◽  
Strong Norm

A smooth soluble abstract linear parabolic equation with the periodic condition on the solution is treated in a separable Hilbert space. This problem is solved approximately by a projection-difference method using the Galerkin method in space and the implicit Euler scheme in time. Effective both in time and in space strong-norm error estimates for approximate solutions, which imply convergence of approximate solutions to the exact solution and order of convergence rate depending of the smoothness of the exact solution, are obtained.

A parallel Crank–Nicolson finite difference method for time-fractional parabolic equation

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
N. H. Sweilam ◽  
H. Moharram ◽  
N. K. Abdel Moniem ◽  
S. Ahmed
Keyword(s):  
Parabolic Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Cluster Computing ◽  
Performance Metrics ◽  
Numerical Test ◽  
Preconditioned Conjugate Gradient ◽  
Pc Cluster ◽  
Crank Nicolson

Abstract- In this paper, a parallel Crank-Nicolson finite difference method (C-N-FDM) for time-fractional parabolic equation on a distributed system using MPI is investigated. The fractional derivative is described in the Caputos sense. The resultant large system of equations is studied using preconditioned conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiment using different number of nodes of the Linux PC cluster is studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.

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
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