To use all functions of this page, please activate cookies in your browser.
With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.
- My watch list
- My saved searches
- My saved topics
- My newsletter
Total variation diminishing
In systems described by partial differential equations, such as the following hyperbolic advection equation,
Additional recommended knowledge
the total variation (TV) is given by,
and the total variation for the discrete case is,
A numerical method is said to be total variation diminishing (TVD) if,
A system is said to be monotonicity preserving if the following properties are maintained as a function of t:
For physically realisable systems where there is energy dissipation of some kind, the total variation does not increase with time. Harten 1986 proved the following properties for a numerical scheme,
Monotone schemes are attractive for solving engineering and scientific problems because they do not provide non-physical solutions.
Godunov's theorem proves that only first order linear schemes preserve monotonicity and are therefore TVD. Higher order linear schemes, whilst more accurate for smooth solutions, are not TVD and tend to introduce spurious oscillations (wiggles) where discontinuities or shocks arise. To overcome these draw backs, various high-resolution, non-linear techniques have been developed, often using flux/slope limiters.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Total_variation_diminishing". A list of authors is available in Wikipedia.|