Changes between Version 33 and Version 34 of u/erica/MusclHancock

Timestamp:

06/18/13 12:12:46 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/MusclHancock

@@ -47 +47 @@
   [[latex($ \vec{u} $)]] 
   in cell i on the nth time level. Using a linear function produces 2 extreme values of each cell, one on the left and on the right (ul and ur). These values are crucial in the MH scheme and their role will be presented shortly. Their values are given by 
-  [[latex($ u_l = u_i(0) = u_i ^n - \frac{1}{2} \triangle_ i $)]]
+
+  [[latex($ u_i ^l = u_i(0) = u_i ^n - \frac{1}{2} \triangle_ i $)]]
+
   and
-   [[latex($ u_r = u_i(0) = u_i ^n + \frac{1}{2} \triangle_ i $)]]
+
+  [[latex($ u_i ^r = u_i(\triangle x) = u_i ^n + \frac{1}{2} \triangle_ i $)]]
    A schematic of the situation is as follows:
 
@@ -58 +61 @@
   [[latex($ \frac{\partial \vec{u}}{\partial t} + \frac{\partial f(\vec{u})}{\partial x} = 0 $)]]
 
+
+  [[latex($ u(x,0) =  \{ ^{u_i^r,~~~~ x <0}_{u_{i+1}^l, ~~x>0} $)]]
  
 
- where 0 is the local origin, i.e. any given intercell boundary.
+ where 0 is the local origin, i.e. any given intercell boundary. We will discuss meaningful ways to set the extreme values, ul and ur, shortly. For now we will just call the data-reconstruction step that which sets the extreme values using the above formula. Given that the expressions for ul and ur involve the slope  [[latex($ \triangle _i$)]], we define 
+
+   [[latex($ \triangle_i = \frac{1}{2}(1+\omega)\triangle u_{i-\frac{1}{2}} + \frac{1}{2}(1-\omega)\triangle u_{i+\frac{1}{2}} $)]]
+
+ where w is a free parameter in the real interval [-1,1]. 
+
+   [[latex($\triangle u_{i-\frac{1}{2}} \equiv u_i^n - u_{i-1}^n$)]]
+
+  [[latex($\triangle u_{i+\frac{1}{2}} \equiv u_{i+1}^n - u_{i}^n$)]]
+
+
+