david/zoltan-discussion-notes

Idea: find sets that are invariant under WENO schemes by considering the smoothness indicator as the defining norm.

Let \(IS(u)\) denote the maximum of the smoothness indicators for the vector \(u\). Can we find a set \(S\) such that \(IS(L(u))\le IS(u)\) for all \(u\in S\), where \(L\) is the WENO scheme (for advection, with some maximum step size).

It may be useful to introduce a different smoothness indicator for this purpose. Or design something WENO-like but different that’s designed to yield set invariance.

We could also just look at how much \(IS(u)\) can grow under an Euler step. Then Zoltan has an approach to proving monotonicity for certain implicit methods.

I made a PyClaw branch that outputs the maximum smoothness indicator at each step: measure_IS (on my workstation only). One needs to run the following:

python advection.py solver_type=sharpclaw kernel_language=Python