Different Limiters

The minmod limiter is very severe. Let’s try a different slope limiter.

A slightly better limiter considers a central difference along with each of the one-sided differences when finding the slope—this is the monotonized-center limiter (or MC limiter):

We start by defining the test for extrema:

\[\xi = (a_{i+1} - a_i) \cdot (a_i - a_{i-1})\]

and then compute the slope as:

\[\begin{split} \left . \Delta a\right |_i = \left \{ \begin{array}{ll} \min \left \{ \frac{1}{2}| a_{i+1} - a_{i-1} |, 2 | a_{i+1} - a_i |, 2 | a_{i} - a_{i-1} | \right \} \mathrm{sign}(a_{i+1} - a_{i-1}) & \xi > 0 \\ 0 & \mathit{otherwise} \end{array}\right . \end{split}\]

Try it

Modify the advection code to use this limiter and compare how well it does on our test problems to the minmod case.