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2.6.2 Monotonicity

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

Definition: Total variation:
TV = \sum_i \mbox{abs} \! \left( \rho_{i+1} - \rho_i \right)
\end{displaymath} (146)
TVD property: A scheme has the TVD property (is ``Total Variation Diminishing'') if \bgroup\color{DEFcolor}$TV$\egroup does not increase in time.

\bgroup\color{DEFcolor}$\Rightarrow$\egroup Non-linear stability criterion.

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

\bgroup\color{DEFcolor}$\Rightarrow$\egroup Try a non-linear scheme.