Changes between Version 13 and Version 14 of u/ehansen/buildcode

Timestamp:

10/31/11 19:19:54 (13 years ago)

Author:

ehansen

Comment:

—

u/ehansen/buildcode

@@ -28 +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))]]
 
-Where [[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:
+Where [[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 (1) with a specific control volume:
 
 [[latex($\int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n+1}) \ \mathrm{d}x = \int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n}) \ \mathrm{d}x + \int_{t^{n}}^{t^{n+1}}F(\tilde{U}(x_{i-\frac{1}{2}},t)) \ \mathrm{d}t - \int_{t^{n}}^{t^{n+1}} F(\tilde{U}(x_{i+\frac{1}{2}},t)) \ \mathrm{d}t \hspace{1 in} (3)$)]] 
@@ -43 +43 @@
 
 [[latex($\tilde{U}(x_{i-\frac{1}{2}},t) = U_{i-\frac{1}{2}}(0) = constant \hspace{1 in} (6)$)]]
+
 [[latex($\tilde{U}(x_{i+\frac{1}{2}},t) = U_{i+\frac{1}{2}}(0) = constant \hspace{1 in} (7)$)]]
 
-Substitute (6) and (7) into (3), and then that result into (2) gives:
+Substitute (6) and (7) into (3), and then substituting that result into (2) gives:
 
 [[latex($U_{i}^{n+1} = \frac{1}{\Delta x} \int^{x_{i+\frac{1}{2}}}_{x_{i - \frac{1}{2}}} \tilde{U}(x,t^{n}) \ \mathrm{d}x + \frac{1}{\Delta x} \int_{t^{n}}^{t^{n+1}}F(U_{i-\frac{1}{2}}(0,t)) \ \mathrm{d}t - \frac{1}{\Delta x} \int_{t^{n}}^{t^{n+1}} F(U_{i+\frac{1}{2}}(0,t)) \ \mathrm{d}t \hspace{1 in} (8)$)]]