Extremum-Preserving Limiters for MUSCL and PPM

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 1 order accurate at all extrema despite the accuracy of the higher-order method [1, 2, 3, 4]. To remedy this problem, higherorder extensions were proposed that relied on elaborate analytic and geometric constructions [5, 6, 7, 8]. 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 2 derivatives. The van Leer limiting can be done separately and implemented in MUSCL (Monotone Upstreamcentered Schemes for Conservation Laws) [2] or done in concert with the parabolic profile limiting and implemented in PPM (Piecewise Parabolic Method) [9, 10]. The extremumpreserving 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 4 order correction to the point-valued, cell-centered initial conditions that is necessary for implementing higher-order limiting. The material herein complements Colella and Sekora [11]. Lastly, there is no guarantee that extremumpreserving limiters preserve positivity. To ensure this property, one should combine the limiting with FCT (Flux-Corrected Transport) [3].