[Paper Review] Extremum-Preserving Limiters for MUSCL and PPM

Michael Sekora, Phillip Colella|ArXiv.org|Mar 24, 2009

Computational Fluid Dynamics and Aerodynamics15 references8 citations

TL;DR

This paper introduces extremum-preserving limiters for MUSCL and PPM schemes that maintain high-order accuracy at smooth extrema by applying nonlinear constraints on second derivatives only at local extrema. The method preserves monotonicity, stability, and $4^{th}$-order spatial accuracy while minimizing computational cost, significantly reducing numerical error in smooth regions without degrading performance at discontinuities.

ABSTRACT

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.

Motivation & Objective

• To address the loss of $4^{\text{th}}$-order accuracy at extrema in traditional limiters for MUSCL and PPM schemes.
• To develop a higher-order limiting strategy that preserves monotonicity and accuracy specifically at smooth extrema without degrading performance at discontinuities.
• To ensure that the limiting process maintains $4^{\text{th}}$-order spatial accuracy in smooth regions while remaining computationally efficient.
• To extend the approach to nonlinear hyperbolic conservation laws using characteristic-based limiting or componentwise application.

Proposed method

• Apply nonlinear constraints on the interpolated values at extrema using combinations of second derivatives to enforce extremum preservation.
• Use van Leer limiting for slope reconstruction in MUSCL and combine it with extremum-preserving correction for $4^{\text{th}}$-order accuracy.
• Implement $4^{\text{th}}$- and $6^{\text{th}}$-order piecewise parabolic reconstruction in PPM, with extremum-preserving limiting applied only at extrema.
• Construct $4^{\text{th}}$-order corrections to cell-centered initial conditions to enable higher-order limiting in both MUSCL and PPM.
• Apply limiters either componentwise to primitive variables or to characteristic fields in nonlinear systems to maintain monotonicity and accuracy.
• Integrate the extremum-preserving limiter into existing MUSCL and PPM frameworks with minimal algorithmic changes, preserving simplicity and efficiency.

Experimental results

Research questions

• RQ1 Can extremum-preserving limiters restore $4^{\text{th}}$-order accuracy at smooth extrema in MUSCL and PPM schemes without sacrificing monotonicity or stability?
• RQ2 How does the performance of extremum-preserving limiters compare to conventional limiters in terms of error reduction for smooth and discontinuous solutions?
• RQ3 What is the sensitivity of the extremum-preserving limiter to the choice of limiting constants $C_{VL}$ and $C_{PPM}$?
• RQ4 Can extremum-preserving limiters be effectively extended to nonlinear hyperbolic conservation laws while maintaining high-order accuracy and monotonicity?

Key findings

• The extremum-preserving limiter reduces $L_1$ error by a factor of ~2.5 for the Gaussian wave (G) at $N=128$ cells, with convergence order $R_1 \approx 2.5$, indicating improved $4^{\text{th}}$-order accuracy at extrema.
• For the semi-circle wave (SC), the limiter reduces $L_1$ error by a factor of ~2.4 at $N=128$, with convergence order $R_1 \approx 1.2$, showing moderate improvement in smooth regions.
• For the square wave (S), the limiter does not reduce $L_1$ error significantly, as expected due to the lack of smooth extrema, with $R_1 \approx 0.8$, consistent with $2^{\text{nd}}$-order behavior.
• The limiter is insensitive to the limiting constants $C_{VL}$ and $C_{PPM}$, with $L_\infty$ error changing by only 4% when $C$ is varied from 1.25 to 5.
• In nonlinear systems, the extremum-preserving limiter produces solutions comparable to WENO3 and WENO5 for the Shu-Osher problem, demonstrating high-resolution performance.
• Numerical results from the Woodward-Colella ramp problem using Chombo with adaptive mesh refinement confirm robustness and stability of the limiter in multidimensional shock-capturing.