Changes between Version 16 and Version 17 of u/johannjc/scratchpad

Timestamp:

01/10/14 16:08:20 (12 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -40 +40 @@
 and the center coefficient as 
 
-[[latex($A_0 =  2 \frac{\kappa \Delta t}{ 2\Delta x^2}T_{i\pm1}^{j^n}  + \rho c_v$)]]
+[[latex($A_0 =  2 \frac{\kappa \Delta t}{ 2\Delta x^2}T_{i}^{j^n}  + \rho c_v$)]]
 
 and the RHS left and right contributions
@@ -46 +46 @@
 [[latex($B_\pm = -\frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i\pm 1}^{j^{n+1}}$)]]
 
-and the RHS source term contribution as 
+and the RHS center term contribution as 
 
-[[latex($B_0 = 2 \frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i\pm 1}^{j^{n+1}} + \rho c_v T_i^j$)]]
+[[latex($B_0 = 2 \frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i}^{j^{n+1}} + \rho c_v T_i^j$)]]
 
 and finally in the code the values for [[latex($A$)]] and [[latex($B$)]] are negated:
@@ -65 +65 @@
 also note that the equation is multipled by the mean molecular weight... which is why [[latex($c_v \rightarrow c_v \times  \chi = \gamma_7$)]] and [[latex($\kappa \rightarrow \kappa \times \chi = \kappa_1$)]]
 
+
+== a note about flux boundary conditions ==
+If we have a specified flux at a boundary - then the equation is just 
+
+[[latex($\rho c_v \frac{\partial T}{\partial t} = -\nabla Q $)]]
+
+which can be discretized as 
+
+[[latex($\rho_i c_v \left (T_i^{j+1}-T_i^j \right ) = -Q_{i+1/2} + Q_{i-1/2} $)]]
+
+which leads to a center coefficient of 
+
+[[latex($A_0 = \rho c_v$)]]
+
+and a left right source term contribution
+
+[[latex($B_\pm = \mp \frac{\Delta t}{\Delta x} Q_{i\pm 1/2}$)]]
+
+and if the left boundary is a flux boundary, but the right side is normal, then we have to adjust the coefficients
+
+[[latex($A_+ = -\frac{\kappa \Delta t}{2 \Delta x^2}T_{i+1}^{j^n} $)]]
+
+and the center coefficient as 
+
+[[latex($A_0 =  \frac{\kappa \Delta t}{ 2\Delta x^2}T_{i}^{j^n}  + \rho c_v$)]]
+
+and the RHS left and right contributions
+
+[[latex($B_+ = -\frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i+1}^{j^{n+1}}$)]]
+
+[[latex($B_-  =  \frac{\Delta t}{\Delta x} Q_{i-1/2}$)]]
+
+and the RHS center term contribution as 
+
+[[latex($B_0 = \frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i}^{j^{n+1}} + \rho c_v T_i^j$)]]
+
+
+Of course - remembering that in AstroBEAR everything is negated...
+
+This is accomplished by
+{{{
+     source = source + ((ndiff-1.0)/(ndiff+1.0))*(T(0)*kx*T(0)**ndiff)
+     source = source - ((ndiff-1.0)/(ndiff+1.0))*(T(p)*kx*T(p)**ndiff)
+     stencil_fixed(0)=stencil_fixed(0)+kx*T(0)**ndiff
+     stencil_fixed(p)=0.0
+     source = source - flb*dt_diff/dx
+}}}