{"title":"High-Order Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws","authors":"Chuan Fan, Kailiang Wu","doi":"arxiv-2409.09632","DOIUrl":null,"url":null,"abstract":"This paper proposes high-order accurate, oscillation-eliminating Hermite\nweighted essentially non-oscillatory (OE-HWENO) finite volume schemes for\nhyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after\neach Runge--Kutta stage, dampening the first-order moments of the HWENO\nsolution to suppress spurious oscillations without any problem-dependent\nparameters. This OE procedure acts as a filter, derived from the solution\noperator of a novel damping equation, solved exactly without discretization. As\na result, the OE-HWENO method remains stable with a normal CFL number, even for\nstrong shocks producing highly stiff damping terms. To ensure the method's\nnon-oscillatory property across varying scales and wave speeds, we design a\nscale- and evolution-invariant damping equation and propose a dimensionless\ntransformation for HWENO reconstruction. The OE-HWENO method offers several\nadvantages over existing HWENO methods: the OE procedure is efficient and easy\nto implement, requiring only simple multiplication of first-order moments; it\npreserves high-order accuracy, local compactness, and spectral properties. The\nnon-intrusive OE procedure can be integrated seamlessly into existing HWENO\ncodes. Finally, we analyze the bound-preserving (BP) property using optimal\ncell average decomposition, relaxing the BP time step-size constraint and\nreducing decomposition points, improving efficiency. Extensive benchmarks\nvalidate the method's accuracy, efficiency, resolution, and robustness.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09632","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
This paper proposes high-order accurate, oscillation-eliminating Hermite
weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for
hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after
each Runge--Kutta stage, dampening the first-order moments of the HWENO
solution to suppress spurious oscillations without any problem-dependent
parameters. This OE procedure acts as a filter, derived from the solution
operator of a novel damping equation, solved exactly without discretization. As
a result, the OE-HWENO method remains stable with a normal CFL number, even for
strong shocks producing highly stiff damping terms. To ensure the method's
non-oscillatory property across varying scales and wave speeds, we design a
scale- and evolution-invariant damping equation and propose a dimensionless
transformation for HWENO reconstruction. The OE-HWENO method offers several
advantages over existing HWENO methods: the OE procedure is efficient and easy
to implement, requiring only simple multiplication of first-order moments; it
preserves high-order accuracy, local compactness, and spectral properties. The
non-intrusive OE procedure can be integrated seamlessly into existing HWENO
codes. Finally, we analyze the bound-preserving (BP) property using optimal
cell average decomposition, relaxing the BP time step-size constraint and
reducing decomposition points, improving efficiency. Extensive benchmarks
validate the method's accuracy, efficiency, resolution, and robustness.