scholarly journals Crank–Nicolson method for solving uncertain heat equation

2022 ◽  
Author(s):  
Jin Liu ◽  
Yifei Hao
Keyword(s):  
Heat Equation ◽  
Crank Nicolson

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

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.

scholarly journals Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 359
Author(s):  
Khudija Bibi ◽  
Tooba Feroze
Keyword(s):  
Finite Difference ◽  
Heat Equation ◽  
Difference Scheme ◽  
Exact Solutions ◽  
Finite Difference Scheme ◽  
Symmetry Transformation ◽  
Discrete Symmetry ◽  
Group Approach ◽  
Comparison Of The Results ◽  
Crank Nicolson

In this article, an invariantized finite difference scheme to find the solution of the heat equation, is developed. The scheme is based on a discrete symmetry transformation. A comparison of the results obtained by the proposed scheme and the Crank Nicolson method is carried out with reference to the exact solutions. It is found that the proposed invariantized scheme for the heat equation improves the efficiency and accuracy of the existing Crank Nicolson method.

scholarly journals A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system

2017 ◽  
Vol 8 (1) ◽  
pp. 43-62
Author(s):  
S. S. Ravindran
Keyword(s):  
Heat Equation ◽  
Mixed Finite Element ◽  
Mhd Equations ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping ◽  
Crank Nicolson

Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics of electro-magnetically and thermally driven flows,involving MHD equations coupled with heat equation. We introduce a partitioned method that allows one to decouplethe MHD equations from the heat equation at each time step and solve them separately. The extrapolated Crank-Nicolson time-stepping scheme is used for time discretizationwhile mixed finite element method is used for spatial discretization. We derive optimal order error estimates in suitable norms without assuming any stability condition or restrictions on the time step size. We prove the unconditional stability of the scheme. Numerical experiments are used to illustrate the theoretical results.

scholarly journals A local Crank-Nicolson method for solving the heat equation

1994 ◽  
Vol 24 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Abdurishit Abuduwali ◽  
Michio Sakakihara ◽  
Hiroshi Niki
Keyword(s):  
Heat Equation ◽  
Crank Nicolson

scholarly journals A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION

2020 ◽  
Vol 5 ◽  
pp. 36
Author(s):  
S. Abbas ◽  
A. A. Khan ◽  
B. Shakia
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Central Difference ◽  
Cyclic Reduction ◽  
Difference Formula ◽  
Reduction Methods ◽  
Crank Nicolson

We study a comparison of serial and parallel solution of 2D-parabolic heat conduction equation using a Crank-Nicolson method with an Alternating Direction Implicit (ADI) scheme. The two-dimensional Heat equation is applied on a thin rectangular aluminum sheet. The forward difference formula is used for time and an averaged second order central difference formula for the derivatives in space to develop the Crank-Nicolson method. FORTRAN serial codes and parallel algorithms using OpenMP are used. Thomas tridigonal algorithm and parallel cyclic reduction methods are employed to solve the tridigonal matrix generated while solving heat equation. This paper emphasize on the run time of both algorithms and their difference. The results are compared and evaluated by creating GNU-plots (Command-line driven graphing utility).

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