Changes between Version 23 and Version 24 of doc/tec/advection

Timestamp:

Nov 12, 2015 2:44:02 PM (9 years ago)

Author:

boeske

Comment:

--

doc/tec/advection

@@ -4 +4 @@
 {{{
 #!Latex
-\[ \frac{\partial \psi}{\partial t} & = & -\frac{\partial ( u \psi )}{\partial x}, \] 
+$\dfrac{\partial \psi}{\partial t} = -\dfrac{\partial ( u \psi )}{\partial x},$
 }}}
 the one dimensional advection equation can be written in the following semidiscrete form: 
 {{{
 #!Latex
-\[ \frac{\partial  \psi_{i} }{\partial t} = - \frac{F_{i+\frac{1}{2}}(u\psi) - F_{i-\frac{1}{2}}(u\psi)}{\Delta x}, \]
+$\dfrac{\partial  \psi_{i} }{\partial t} = - \dfrac{F_{i+\frac{1}{2}}(u\psi) - F_{i-\frac{1}{2}}(u\psi)}{\Delta x},$
 }}}
 where 
 {{{
 #!Latex
-\[F_{i\pm\frac{1}{2}} \]
+$F_{i\pm\frac{1}{2}}$
 }}}
 denotes the fluxes staggered half a grid length related to the advected quantity. \\
@@ -20 +20 @@
 {{{
 #!Latex
-\[ F_{i-\frac{1}{2}}^{6} &=& \frac{u_{i-\frac{1}{2}}}{60} \left[ 37(\psi_{i}+\psi_{i-1}) - 8(\psi_{i+1} + \psi_{i-2}) +(\psi_{i+2} + \psi_{i-3}) \right] \] 
-}}}
-{{{
-#!Latex
-\[ F_{i-\frac{1}{2}}^{5} &=& F_{i-\frac{1}{2}}^{6} - \frac{|u_{i-\frac{1}{2}}|}{60} \left[10(\psi_{i}-\psi_{i-1}) -5(\psi_{i+1} - \psi_{i-2})+(\psi_{i+2} - \psi_{i-3}) \right] . \]
+$F_{i-\frac{1}{2}}^{6} = \frac{u_{i-\frac{1}{2}}}{60} \left[ 37(\psi_{i}+\psi_{i-1}) - 8(\psi_{i+1} + \psi_{i-2}) +(\psi_{i+2} + \psi_{i-3}) \right] ,$
+}}}
+ 
+{{{
+#!Latex
+$F_{i-\frac{1}{2}}^{5} = F_{i-\frac{1}{2}}^{6} - \dfrac{|u_{i-\frac{1}{2}}|}{60} \left[10(\psi_{i}-\psi_{i-1}) -5(\psi_{i+1} - \psi_{i-2})+(\psi_{i+2} - \psi_{i-3}) \right] . $
 }}}
 The 5^th^ order upwind discretization (WS5) consists of a centered non dissipative 6^th^ (WS6) order flux and an artificially added numerical dissipation term. This term is necessary to stabilize the numerical solution, because higher order centered fluxes exhibits worse stability properties. The absolute value of the advective velocity component in the dissipation term removes a sign-dependent effect of the velocity and assures a dissipative effect also for u < 0. 
@@ -33 +34 @@
 {{{
 #!Latex
-\[ \frac{\partial \hat{\psi}_{\kappa}}{\partial t}  =  - \frac{i}{\Delta t} C_{r} \kappa_{eff}\,\hat{\psi}_{\kappa}, \]
+$\dfrac{\partial \hat{\psi}_{\kappa}}{\partial t}  =  - \dfrac{i}{\Delta t} C_{r} \kappa_{eff}\,\hat{\psi}_{\kappa},$
 }}}
 where
 {{{
 #!Latex
-\[
-\hat{\psi}  
+$\hat{\psi}$ 
 }}}
 denotes the fourier transformed of ψ. The Courant number
 {{{
 #!Latex
-\[ 
-C_{r} = \frac{u \Delta t}{\Delta x} 
-\]
+$C_{r} = \dfrac{u \Delta t}{\Delta x}$
 }}}  
 characterizes stability properties and i is the imaginary unit.