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Fifth order anti diffusive scheme for 1D hyperbolic heat transferHello
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: 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 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 ShuFrom 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: http://img109.imageshack.us/img109/3063/17886845.jpg Where the following equation of characteristic variables will be obtained: http://img135.imageshack.us/img135/5480/86947594.jpg http://img831.imageshack.us/img831/8894/72769500.jpg 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: http://img89.imageshack.us/img89/6877/85513480.jpg Where http://img840.imageshack.us/img840/9501/1and2.jpg 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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