Transport and averaging are easy and well determined. The entire
algorithm is determined by the
- Consistency: ``reasonable'' interpolation
- Accuracy: high-order polynomials
- Conservativity: proper flux formulation
- Stability, positivity:
A scheme has the TVD property
(is ``Total Variation Diminishing'')
does not increase in time.
Non-linear stability criterion.
Godunov's theorem: linear monotonicity-preserving methods are first-order accurate, at best.
Try a non-linear scheme.