Changes between Version 46 and Version 47 of ModulesOnAstroBear

Timestamp:

04/20/17 12:52:56 (8 years ago)

Author:

Jonathan

Comment:

—

ModulesOnAstroBear

@@ -101 +101 @@
 Anything that is defined in the {{{InfoDef}}} type is now available.  For example, {{{q}}} need not be defined...just reference it by {{{Info%q}}}.
 
-There are two major data arrays in {{{InfoDef}}}: the {{{q}}} array and the {{{aux}}} array.  {{{q}}} holds the volume averaged data and is used by all AstroBEAR simulations while {{{aux}}} holds face averaged data and is used only for MHD when nDim > 1.  Note that volume averaged data and cell-centered data are often used interchangeably, but there is an important distinction.  Take for instance a simple function f(x) defined on the interval [0:h].  The average of f(x) over the interval is 
-
- f(0)+d,,1,,f(0)h/2+d,,2,,f(0)h^2^/6+... 
-
-where the cell centered value is
-
- f(0)+d,,1,,f(0)h/2+d,,2,,f(0)h^2^/8+... 
+There are two major data arrays in {{{InfoDef}}}: the {{{q}}} array and the {{{aux}}} array.  {{{q}}} holds the volume averaged data and is used by all AstroBEAR simulations while {{{aux}}} holds face averaged data and is used only for MHD when nDim > 1.  Note that volume averaged data and cell-centered data are often used interchangeably, but there is an important distinction.  Take for instance a simple function $f(x)$ defined on the interval $[-h/2:h/2]$.  Applying a Taylor expansion to $f$ and finding the average of $f(x)$ gives
+
+$\bar{f}=f(0)+\frac{1}{24} f''(0) h^2 + ...$
+
+where the cell centered value is $f(0)$
 
 so the cell centered value is second order accurate for the volume average and usually is a quick way to estimate the volume average.  However if the function has large 2nd derivatives (or higher) this can lead to large errors in the volume average.  This is often apparent when modeling discontinuities along curved boundaries.  There are two ways to handle this problem: