TVD scheme

From CFD-Wiki

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

TVD condition: Incremental form

TVD condition: Viscosity form

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TVD scheme

From CFD-Wiki

TVD condition: Incremental form

TVD condition: Viscosity form

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