Changes between Version 7 and Version 8 of u/ehansen/buildcode


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Timestamp:
10/31/11 18:15:33 (13 years ago)
Author:
ehansen
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  • u/ehansen/buildcode

    v7 v8  
    1414The Euler equations can also be written in integral form for a general control volume  [[latex($[x_1,x_2] \ \mathrm{x} \ [t_1,t_2]$)]] .  To simplify the notation, let U be a column matrix containing the conserved quantities and F be the matrix of the corresponding fluxes. Now the Euler equations are:
    1515
    16  [[latex($\int^{x_2}_{x_1} U(x,t_2) \ \mathrm{d}x = \int_{x_1}^{x_2} U(x,t_1) \ \mathrm{d}x + \int_{t_1}^{t_2}F(x_1,t) \ \mathrm{d}t - \int_{t_1}^{t_2} F(x_2,t) \ \mathrm{d}t \ \ \ \ \ (1)$)]] 
     16 [[latex($\int^{x_2}_{x_1} U(x,t_2) \ \mathrm{d}x = \int_{x_1}^{x_2} U(x,t_1) \ \mathrm{d}x + \int_{t_1}^{t_2}F(x_1,t) \ \mathrm{d}t - \int_{t_1}^{t_2} F(x_2,t) \ \mathrm{d}t \hspace{1 in} (1)$)]] 
    1717
    1818In this form, it might be easier to see that the change in conserved quantity is equal to the flux coming into the volume minus the flux going out of the volume.
     
    2626First, we want to use the integral form of the Euler equations (1).  Next, we define a cell average for cell i at time step n + 1: 
    2727
    28 [[latex($U_{i}^{n+1} = \frac{1}{\Delta x} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \tilde{U}(x,t^{n+1}) \ \mathrm{d}x$ \ \ \ \ \ (2))]]
     28[[latex($U_{i}^{n+1} = \frac{1}{\Delta x} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \tilde{U}(x,t^{n+1}) \ \mathrm{d}x$ \hspace{1 in}(2))]]
    2929
    3030Where [[latex($\tilde{U}$)]] is the global solution as opposed to the local solution [[latex($U$)]].  However, we already know that the global solution can be written in terms of its fluxes.  You just use equation (1) with a specific control volume:
    3131
    32 [[latex($\int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n+1}) \ \mathrm{d}x = \int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n}) \ \mathrm{d}x \ + \ \int_{t^{n}}^{t^{n+1}}F(x_{i-\frac{1}{2}},t) \ \mathrm{d}t \ - \ \int_{t^{n}}^{t^{n+1}} F(x_{i+\frac{1}{2}},t) \ \mathrm{d}t \ \ \ \ \ (3)$)]]
     32[[latex($\int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n+1}) \ \mathrm{d}x = \int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n}) \ \mathrm{d}x \ + \ \int_{t^{n}}^{t^{n+1}}F(x_{i-\frac{1}{2}},t) \ \mathrm{d}t \ - \ \int_{t^{n}}^{t^{n+1}} F(x_{i+\frac{1}{2}},t) \ \mathrm{d}t \hspace{1 in} (3)$)]]
    3333
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