Total Variation Diminishing Schemes
High-resolution schemes that guarantee no new oscillations by ensuring the solution's total variation never increases.
A principled route to oscillation-free accuracy
For scalar conservation laws the exact solution never develops new oscillations: its total variation, the sum of the sizes of all its ups and downs, does not increase over time. A numerical scheme that respects this property is called total variation diminishing (TVD). A TVD scheme cannot create the spurious wiggles near shocks that plague high-order linear schemes, so TVD is a rigorous design principle for shock-capturing methods.
Godunov's theorem forces nonlinearity
Godunov proved that any linear scheme that preserves monotonicity can be at most first-order accurate. To be both high-order and oscillation-free, a scheme must be nonlinear, adapting its behavior to the local solution. TVD schemes achieve this through flux or slope limiters that depend on the data, blending high-order accuracy in smooth regions with first-order robustness near discontinuities.
Flux limiters
A flux limiter examines the ratio of consecutive solution gradients and scales the high-order correction accordingly. Where the solution is smooth (gradient ratio near one) it allows the full high-order flux; near an extremum or shock (gradient ratio negative or large) it reduces toward the safe first-order flux. Popular limiters include minmod (most diffusive), superbee (sharpest), van Leer, and MC, each occupying a known region of the TVD stability diagram.
- minmod: safe and diffusive
- van Leer: smooth and widely used
- superbee: sharpest but can steepen smooth profiles
- MC: a balanced compromise
Context and limits
TVD schemes are second-order accurate in smooth regions but drop to first order at smooth extrema, a limitation that motivated the higher-order WENO family. The TVD concept rigorously applies only to scalar one-dimensional problems, but limiters extend heuristically and effectively to systems and multiple dimensions, where they stabilize the finite volume transport of density, momentum, and magnetic flux in plasma fluid codes.