Modified Crank–Nicolson Scheme with Richardson Extrapolation for One-Dimensional Heat Equation

AbstractThe numerical solution of a parabolic Volterra integro-differential equation with a memory term on a one-dimensional unbounded spatial domain is considered. A quasi-wavelet based numerical method is proposed to handle the spatial discretisation, the Crank-Nicolson scheme is used for the time discretisation, and second-order quadrature to approximate the integral term. Some numerical examples are presented to illustrate the efficiency and accuracy of this approach.

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Second-Order Convergence and Unconditional Stability on Crank-Nicolson Scheme for Burgers’ Equation

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.871.15 ◽

2013 ◽

Vol 871 ◽

pp. 15-20

Author(s):

Quan Zheng ◽

Lei Fan ◽

Guan Ying Sun

Keyword(s):

Boundary Conditions ◽

Heat Equation ◽

Burgers Equation ◽

Second Order ◽

Dirichlet Boundary Conditions ◽

Second Order Convergence ◽

Nicolson Scheme ◽

Homogeneous Dirichlet Boundary Conditions ◽

Robin Boundary ◽

Crank Nicolson

In this paper, we study the numerical solution of one-dimensional Burgers equation with non-homogeneous Dirichlet boundary conditions. This nonlinear problem is converted into the linear heat equation with non-homogeneous Robin boundary conditions by Hopf-Cole transformation. The heat equation is discretized by Crank-Nicolson finite difference scheme, and the fourth-order difference schemes for the Robin conditions are combined with the Crank-Nicolson scheme at two endpoints. The proposed method is proved to be second-order convergent and unconditionally stable. The numerical example supports the theoretical results.

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Incident wave and unidirectional wave source implementation in Crank-Nicolson scheme for one-dimensional Maxwell's equations

2017 Progress in Electromagnetics Research Symposium - Fall (PIERS - FALL) ◽

10.1109/piers-fall.2017.8293653 ◽

2017 ◽

Author(s):

Satyajeet Singh ◽

Harishankar Ramachandran

Keyword(s):

Incident Wave ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Wave Source ◽

One Dimensional ◽

Nicolson Scheme ◽

Crank Nicolson

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

International Journal of Differential Equations ◽

10.1155/2011/520840 ◽

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.

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