# TVD scheme

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 Revision as of 04:03, 30 September 2005 (view source)Praveen (Talk | contribs)← Older edit Latest revision as of 21:28, 15 October 2005 (view source)Jola (Talk | contribs) (2 intermediate revisions not shown) Line 3: Line 3: :$:[itex] TV(u^{n+1}) \le TV(u^n) TV(u^{n+1}) \le TV(u^n) +$ + + The total variation of a grid function is defined as + + :$+ TV(u) = \sum_j | u_{j+1} - u_j|$ [/itex] Note that a TVD scheme may not satisfy the entropy condition and hence can give incorrect solution. We have the following relationship between monotone, TVD and monotonicity preserving schemes, Note that a TVD scheme may not satisfy the entropy condition and hence can give incorrect solution. We have the following relationship between monotone, TVD and monotonicity preserving schemes, - [[Monotone scheme]] $\Longrightarrow$ [[TVD scheme]] $\Longrightarrow$ [[Monotonicity preserving scheme]] + : [[Monotone scheme]] $\Longrightarrow$ [[TVD scheme]] $\Longrightarrow$ [[Monotonicity preserving scheme]] ==TVD condition: Incremental form== ==TVD condition: Incremental form== ==TVD condition: Viscosity form== ==TVD condition: Viscosity form== + + {{Stub}}

## Latest revision as of 21:28, 15 October 2005

A scheme is said to be TVD or Total Variation Diminishing if it does not increase the total variation of the solution, i.e.,

$TV(u^{n+1}) \le TV(u^n)$

The total variation of a grid function is defined as

$TV(u) = \sum_j | u_{j+1} - u_j|$

Note that a TVD scheme may not satisfy the entropy condition and hence can give incorrect solution. We have the following relationship between monotone, TVD and monotonicity preserving schemes,

Monotone scheme $\Longrightarrow$ TVD scheme $\Longrightarrow$ Monotonicity preserving scheme