Total variation diminishing

Property of discretization schemes used to solve hyperbolic partial differential equations
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 Computational Fluid Dynamics, TVD scheme is employed to capture sharper shock predictions without any misleading oscillations when variation of field variable “” is discontinuous. To capture the variation fine grids ( 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.

Monotone schemes are attractive for solving engineering and scientific problems because they do not produce non physical solutions. Godunov's theorem proves that linear schemes which preserve monotonicity are, at most, only first order accurate. Higher order linear schemes, although more accurate for smooth solutions, are not TVD and tend to introduce spurious oscillations (wiggles) where discontinuities or shocks arise. To overcome these drawbacks, various high resolution, non linear techniques have been developed, often using flux/slope limiters.