Local maximum principle satisfying high‐order non‐oscillatory schemes‡. (9th December 2015)

• Record Type: Journal Article
• Authors: Dubey, Ritesh Kumar; Biswas, Biswarup; Gupta, Vikas
• Is Part Of: International journal for numerical methods in fluids. Volume 81:Number 11(2016)
• Page Start: 689
• Page End: 715
• Publication Date: 2015-12-09
• Subjects: hyperbolic conservation laws -- smoothness parameter -- non‐sonic critical point -- total variation stability -- finite difference schemes; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4202
• Languages: English
• ISSNs: 0271-2091

Abstract

Summary: The main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd. Abstract : Using a local maximum principle, the solution region of hyperbolic scalar conservation law is classified into sub‐regions where at least second‐order non‐oscillatory approximation can be achieved. Nonlinear stability bounds are given, which ensure for non‐occurrence of induced oscillations by second‐order schemes. Using these bounds, second‐order accurate hybrid numerical schemes are constructed with the help of a shock detector, which can preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error.