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TVD scheme

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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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