  Limiters are nonlinear hybridization techniques that are used to preserve
positivity and monotonicity when numerically solving hyperbolic conservation
laws. Unfortunately, the original methods suffer from the truncation-error
being first-order accurate at all extrema despite the accuracy of the
higher-order method. To remedy this problem, higher-order extensions were
proposed that relied on elaborate analytic and geometric constructions. Since
extremum-preserving limiters are applied only at extrema, additional
computational cost is negligible. Therefore, extremum-preserving limiters
ensure higher-order spatial accuracy while maintaining simplicity. This report
presents higher-order limiting for (i) computing van Leer slopes and (ii)
adjusting parabolic profiles. This limiting preserves monotonicity and accuracy
at smooth extrema, maintains stability in the presence of discontinuities and
under-resolved gradients, and is based on constraining the interpolated values
at extrema (and only at extrema) by using nonlinear combinations of second
derivatives. The van Leer limiting can be done separately and implemented in
MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) or done in
concert with the parabolic profile limiting and implemented in PPM (Piecewise
Parabolic Method). The extremum-preserving limiters elegantly fit into any
algorithm which uses conventional limiting techniques. Limiters are outlined
for scalar advection and nonlinear systems of conservation laws. This report
also discusses the fourth-order correction to the point-valued, cell-centered
initial conditions that is necessary for implementing higher-order limiting.
