Changes between Version 28 and Version 29 of u/bliu/pnfldiff

Timestamp:

08/16/18 16:50:23 (6 years ago)

Author:

Baowei Liu

Comment:

—

u/bliu/pnfldiff

@@ -69 +69 @@
 Discretize the equation of $n_{e}$ in the same way as [https://astrobear.pas.rochester.edu/trac/wiki/ThermalConduction Thermal Conduction]
 
-$\partial_{t}n_{e} +\partial_{i}(n_{e}u_{i})=I-r$
+$\partial_{t}n_{e} +\partial_{i}(n_{e}u_{i})=\partial_{i}J_{i}-r$
 
 Replace $n_{e}$ with $n_{e}^{*}$ where[[BR]]
@@ -76 +76 @@
 $\Phi=1$ for backward Euler, $\Phi=1/2$ for Crank Nicholson
 
-$\partial_{t}n_{e} +u_{i}\partial_{i}n_{e}^{*}+n_{e}^{*}\partial_{i}u_{i}=I-r$
+$\partial_{t}n_{e} +u_{i}\partial_{i}n_{e}^{*}+n_{e}^{*}\partial_{i}u_{i}=\partial_{i}J_{i}-r$
 
-
-
+$\partial_{t}n_{e}=\frac{n_{e}^{\prime\hat{i}}-n_{e}^{\hat{i}}}{\Delta t} $
+$-(1-\Phi)u_{i}\partial_{i}n_{e}=(1-\Phi)u_{\hat{i}}\frac{n^{\hat{i}+1}_{e}-n^{\hat{i}-1}_{e}}{2\Delta x}$
+$-\Phi u_{i}\partial_{i}n^{'}_{e}=-\Phi u_{\hat{i}}\frac{n^{'\hat{i}+1}_{e}-n^{'\hat{i}-1}_{e}}{2\Delta x}$
+$-(1-\Phi)n_{e}\partial_{i}u_{i}=(1-\Phi)n_{e}^{\hat{i}}\frac{u_{\hat{i}+1}-u_{\hat{i}-1}}{2\Delta x}$
+$-\Phi n_{e}^{'}\partial_{i}u_{i}=-\Phi\frac{u_{\hat{i}+1}-u_{\hat{i}-1}}{2\Delta x}$
 
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