Total variation diminishing

In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten.

In systems described by partial differential equations, such as the following hyperbolic advection equation,

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 numerical scheme is said to be monotonicity preserving if the following properties are maintained:

Harten 1983 proved the following properties for a numerical scheme,

In Computational Fluid Dynamics, TVD scheme is employed to capture sharper shock predictions without any misleading oscillations when variation of field variable “Ø” is discontinuous. To capture the variation fine grids (∆x = very small) are needed and the computation becomes heavy and therefore uneconomic. The use of coarse grids with central difference scheme, upwind scheme, hybrid difference scheme, and power law scheme gives false shock predictions. TVD scheme enables sharper shock predictions on coarse grids saving computation time and as the scheme preserves monotonicity there are no spurious oscillations in the solution.

Consider the steady state one-dimensional convection diffusion equation,

where ${\displaystyle \rho }$ is the density, ${\displaystyle \mathbf {u} }$ is the velocity vector, ${\displaystyle \phi }$ is the property being transported, ${\displaystyle \Gamma }$ is the coefficient of diffusion and ${\displaystyle S_{\phi }}$ is the source term responsible for generation of the property ${\displaystyle \phi }$.

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