Changes between Version 2 and Version 3 of ThermalConduction

Timestamp:

01/16/14 11:27:05 (11 years ago)

Author:

Baowei Liu

Comment:

—

ThermalConduction

@@ -134 +134 @@
 }}}
 
+== If the right boundary is a zero-slope boundary, we have ==
+[[latex($T_{i+1}^{j+1}=T_{i}^{j+1}$)]]
+and 
+[[latex($T_{i+1}^{j}=T_{i}^{j}$)]]
+
+So now on the right boundary, the equation becomes
+{{{
+#!latex
+\setcounter{equation} 4
+\begin{eqnarray}
+\rho_i c_v \left ( T_i^{j+1}-T_i^j \right ) & =  & \frac{\kappa \Delta t}{2 \Delta x^2}\left (- T_{i}^{j^n}  T_{i}^{j+1} + T_{i-1}^{j^n}  T_{i-1}^{j+1} \right ) \\
+ & - & \frac{\left (n - 1 \right ) \kappa \Delta t}{ 2 \left ( n+ 1 \right ) \Delta x^2} \left (-  T_{i}^{j^{n+1}} + T_{i-1}^{j^{n+1}} \right )  
+\end{eqnarray}
+}}}
+And the coeeficients becomes
+
+[[latex($A_+ = 0 $)]]
+
+[[latex($A_-=-\frac{\kappa \Delta t}{2 \Delta x^2}T_{i-1}^{j^n} $)]]
+
+[[latex($A_0=\frac{\kappa \Delta t}{2 \Delta x^2}T_{i}^{j^n}T_{i}^{j+1}+ \rho c_v$)]]
+
+and the RHS left and right contributions
+
+[[latex($B_+=0 $)]]
+
+[[latex($B_-=-\frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i\pm 1}^{j^{n+1}}$)]]
+
+[[latex($B_0=\frac{\left (n - 1 \right ) \kappa \Delta t}{ 2  \left ( n+ 1 \right ) \Delta x^2} T_{i}^{j^n}+\rho C_vT_{i}^{j}$)]]
+
+Again in AstroBEAR everything is negated
+
+This is accomplished by
+
+{{{
+stencil_fixed(0) =stencil_fixed(0)+kx*info%q(iq(1),iq(2),iq(3),itemp)**ndiff ! store the stencil value to temp
+!source = source -((ndiff-1.0)/(ndiff+1.0))*(stencil_fixed(p)*info%q(ip(1),ip(2),ip(3),itemp)-kx*info%q(iq(1),iq(2),iq(3),itemp)**(ndiff+1))
+source = source -((ndiff-1.0)/(ndiff+1.0))*(stencil_fixed(p)*info%q(ip(1),ip(2),ip(3),itemp)+kx*info%q(iq(1),iq(2),iq(3),itemp)**(ndiff+1))
+stencil_fixed(p) = 0.0 ! zero out the stencil
+}}}
+
+where the commented out line is the old-version which got the sign before the "kx" wrong.
+
 == Limiting case ==