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Fifth order anti diffusive scheme for 1D hyperbolic heat transfer 

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November 3, 2010, 05:48 
Fifth order anti diffusive scheme for 1D hyperbolic heat transfer

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Recently, Wensheng Shen et al published a paper as "antidiffusive methods for hyperbolic heat transfer" that consider WENO as one of the antidiffusive methods. Link to download this paper: http://www.cps.brockport.edu/~shen/p...iDiffusive.pdf I am trying to review this paper and want to solve 1D hyperbolic heat transfer equation using fifthorder 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 “antidiffusive 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: Where 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. Last edited by areffallah; November 3, 2010 at 06:04. 

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