Minmod

Minmod is a nonlinear slope limiter used in numerical methods for hyperbolic conservation laws to suppress spurious oscillations near discontinuities while preserving monotonicity. It is a simple limiter that appears in high-resolution schemes such as MUSCL and TVD (total-variation-diminishing) formulations.

Mathematically, for real numbers a and b, the two-argument minmod is defined as minmod(a,b) = a if a and b have the same sign and |a| <= |b|; minmod(a,b) = b if a and b have the same sign and |b| < |a|; and minmod(a,b) = 0 if a and b have opposite signs (or if either is zero). Equivalently, minmod(a,b) = sgn(a) * min(|a|,|b|) when a*b > 0, and 0 otherwise. The concept extends to multiple arguments: minmod(a1,...,an) returns the common-signed value with the smallest magnitude if all ai share the same sign; otherwise it returns 0.

In practice, minmod is used to limit the slope in the reconstruction of interface values in finite-volume or finite-difference discretizations. This helps enforce the TVD property for certain schemes and prevents the formation of new extrema near shocks or steep gradients. Compared with other limiters, minmod is relatively diffusive and tends to smear sharp features, but it offers robustness and simplicity. It is often employed as a building block in more complex limiters and as a benchmark for monotone reconstruction.

Related concepts include other slope limiters such as van Leer, Superbee, and the Monotonized Central (MC) limiter. The minmod function remains a widely used tool in numerical methods for hyperbolic problems.

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