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areffallah November 3, 2010 06:48

Fifth order anti diffusive scheme for 1D hyperbolic heat transfer

Recently, Wensheng Shen et al published a paper as "anti-diffusive methods for hyperbolic heat transfer" that consider WENO as one of the anti-diffusive methods.

Link to download this paper:

I am trying to review this paper and want to solve 1D hyperbolic heat transfer equation using fifth-order WENO. I have studied some papers about applying WENO, as listed below, But they couldn't solve my problem.
[1] A finite volume high order ENO scheme for two dimensional hyperbolic systems. By: Casper and Atkins
[2] Efficient Implementation of Weighted ENO Schemes. By: Jiang and Shu
[3] Essentially non_oscillatory and weighted essentially non_oscillatory schemes for hyperbolic conservation laws. By: Chi_Wang Shu
From This Paper I mean “anti-diffusive methods for hyperbolic heat transfer”.

I have two basic problems:
Considering This Paper, the corresponding mathematical model consists of the following two equations:
Where the following equation of characteristic variables will be obtained:
1) I want to apply boundary and initial conditions as below:
The dimensionless temperature of the slab is kept at T = 0 initially, and the temperature at the left boundary is increased to T = 1 at time t > 0. Two types boundary conditions are considered at the right end of the slab, given temperature (T = 0) and zero heat flux (q = 0).
Main problem is: How should I apply boundary and initial conditions. Because I can only specify either T or q but not both at boundaries. For initial conditions, I have only T and for left and right boundary conditions I have only one of T or q, therefore what should I do for another variable value? T and q or W1 and W2.
W1 = 0.5*(T + q) and W2 = 0.5*(T - q)

2) As explained in procedure 2.9. [3], I got the procedure for solving this problem as below:
At each fixed X(i+1/2)
1) Compute the right eigenvectors the left eigenvectors and the eigenvalues.
2) Using left eigenvectors, transform [T q] vector to [W1 W2].
3) Perform weno reconstruction procedure on W1 and W2 to obtain W1(i+1/2) and W2(i+1/2).
4) Transform back from characteristic to physical domain using right eigenvectors:
[T(i+1/2) q(i+1/2)] = R*[W1(i+1/2) W2(i+1/2)]
Where : R = right eigenvectors and Inv(R) = left eigenvectors.
5) Form the scheme below:
And I consider two equations separately to solve (in this step I mean).

Is this procedure True? If not, please tell me what is my fault here.

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