Total Variation Diminishing Schemes
TVD schemes guarantee that the total variation of the solution does not increase, preventing the growth of spurious oscillations.
A monotonicity guarantee
The total variation of a discrete solution is the sum of the absolute differences between neighboring values, a measure of how much the solution wiggles. A total variation diminishing (TVD) scheme is one for which this quantity never increases from one time step to the next. Since physical solutions of scalar conservation laws do not grow their total variation, a TVD scheme cannot manufacture new oscillations, overshoots, or undershoots.
This property is central to robust shock and front capturing: it rules out the ringing that plagues naive high-order schemes near discontinuities, while still permitting sharp resolution of those discontinuities.
Godunov's order barrier
Godunov's theorem states that any linear monotone scheme can be at most first-order accurate. TVD schemes escape this barrier by being nonlinear even when applied to linear equations: the flux limiter makes the coefficients depend on the solution itself. This nonlinearity is what allows second-order accuracy in smooth regions together with monotone behavior at discontinuities.
Construction and reach
Second-order TVD schemes are built by adding a limited anti-diffusive flux to a first-order upwind base, with the limiter constrained to Sweby's admissible region. The framework is rigorous for one-dimensional scalar problems; for systems and multiple dimensions TVD is applied component- or direction-wise and holds approximately rather than exactly, but still delivers robust, essentially non-oscillatory results.
- Total variation never increases in time
- Prevents spurious new extrema near discontinuities
- Godunov barrier requires nonlinearity for high order
- Extended dimensionally and to systems in practice
TVD schemes and their WENO successors are the standard for capturing sharp gradients in fusion fluid and MHD simulations without corrupting the surrounding solution.