Extremum-Preserving Limiters for MUSCL and PPM

Michael Sekora, Princeton University

Limiters are nonlinear hybridization techniques that are used to preserve positivity and monotonicity when numerically solving hyperbolic conservation laws such as the Euler equations in fluid dynamics. Unfortunately, the original limiting methods used in MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) and PPM (Piecewise Parabolic Method) suffer from the truncation error being 1st order accurate at all extrema despite the overall accuracy of the higher-order method. To remedy this problem, higher-order extensions such as the WENO (Weighted Essentially Non-Oscillatory) Method were developed which relied on elaborate analytic and/or geometric constructions. In revisiting the limiter problem, Colella and Sekora derived a new family of limiting techniques that ensure higher-order spatial accuracy while maintaining simplicity such that the extremum-preserving limiters directly plug into any algorithm which uses conventional limiting techniques. The Colella-Sekora limiters are based on constraining interpolated values at extrema (and only at extrema) and using nonlinear combinations of various difference approximations of the second derivatives.

Abstract Author(s): Michael D Sekora