Changes between Version 11 and Version 12 of u/erica/MusclHancock

Timestamp:

06/17/13 13:59:08 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/MusclHancock

@@ -1 @@
-= Theory = 
+= Introduction = 
 
 The Muscl-Hancock scheme is a 2nd order numerical method. The higher the order of accuracy, the more convergent is the scheme. For instance, here is a plot of the Godunov 1st order upwind scheme (aka CIR scheme, 1st order monotone, what I have been using), the Lax Friedrichs scheme (1st order monotone), Lax Wendroff scheme (2nd order, non-monotone), and Warming-Beam (2nd order, non-monotone).
@@ -16 +16 @@
 [[Image(compare3.png,35%)]]
 
-Unfortunately, all monotone schemes are 1st order at best, so the challenge is to reduce these oscillations in regions of discontinuous flow while maximizing accuracy in regions of smooth flow. One such class of schemes are known as 'Total Variation Diminishing' (TVD). These schemes implicitly add artificial viscosity, resulting in smoother flows near large gradients. 
+Note the ugly, spurious oscillations near the large gradients of the step function. Unfortunately, all monotone schemes (those that do not contain artificial oscillations) are 1st order at best. Thus, the challenge is to reduce these oscillations in regions of discontinuous flow while maximizing accuracy in regions of smooth flow. One such class of schemes are known as 'Total Variation Diminishing' (TVD). These schemes implicitly add artificial viscosity, resulting in smoother flows near large gradients. Such a scheme is the Muscl-Hancock scheme.
 
-= Steps =
+= Steps in the Muscl Hancock (MH) Scheme =
+
+To produce 2nd-order accuracy, an outline of the MH scheme is as follows.
+
+At this point, we are left with a higher order accurate code, but not one that is free of spurious oscillations near large gradients. To circumvent this, we need to add a TVD measure.
 
 = Algorithm Outline =