MUSCL和PPM的保压限位器
Extremum-Preserving Limiters for MUSCL and PPM
摘要：
限制器是非线性杂交技术，用于在数值求解双曲守恒律时保持正性和单调性。不幸的是，尽管高阶方法的精度，但原始方法的截断误差在所有极值处都是一阶精度。为了解决这个问题，高阶扩展被提出，依赖于精心设计的分析和几何结构。由于极值保持限制器仅在极值处应用，因此额外的计算成本可以忽略不计。因此，极值保持限制器确保高阶空间精度，同时保持简单性。这份报告提出了高阶限制（i）计算范里尔斜率和（ii）调整抛物线轮廓。这种限制在平滑极值处保持单调性和准确性，在存在不连续性和欠分辨梯度的情况下保持稳定性，并且基于通过使用二阶导数的非线性组合来约束极值处（并且仅在极值处）的插值。van Leer限制可以单独完成并在MUSCL（Monotone Upstream-centered Schemes for Conservation Laws）中实现，或者与抛物线轮廓限制一起完成并在PPM（分段抛物线方法）中实现。极值保持限制器优雅地适合任何使用传统限制技术的算法。限制器概述了标量平流和非线性系统的守恒律。本报告还讨论了四阶校正的点值，细胞为中心的初始条件是必要的实施高阶限制。
英文摘要：
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.
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