2.6.2 Monotonicity

Transport and averaging are easy and well determined. The entire algorithm is determined by the reconstruction scheme.

• Consistency: ``reasonable'' interpolation
• Accuracy: high-order polynomials
• Conservativity: proper flux formulation
• Stability, positivity: monotonicity preservation

Definition: Total variation:

$$TV = \sum_i \mbox{abs} \! \left( \rho_{i+1} - \rho_i \right)$$ (146)

TVD property: A scheme has the TVD property (is ``Total Variation Diminishing'') if $TV$ does not increase in time.

$\Rightarrow$ Non-linear stability criterion.

Godunov's theorem: linear monotonicity-preserving methods are first-order accurate, at best.

$\Rightarrow$ Try a non-linear scheme.