On sub-linear convergence for linearly degenerate waves in capturing schemes
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| Title: | On sub-linear convergence for linearly degenerate waves in capturing schemes |
|---|---|
| Authors: | Banks, J.W.1,2 banksj@alum.rpi.edu, Aslam, T.3, Rider, W.J.2 |
| Source: | Journal of Computational Physics. Jul2008, Vol. 227 Issue 14, p6985-7002. 18p. |
| Subjects: | Shock waves, Mechanical shock, Differential equations, Fluid dynamics |
| Abstract: | Abstract: A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of Δt 1/(p+1) which yields an estimate for the convergence rate as p/(p +1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations. [Copyright &y& Elsevier] |
| Copyright of Journal of Computational Physics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 32560808 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: On sub-linear convergence for linearly degenerate waves in capturing schemes – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Banks%2C+J%2EW%2E%22">Banks, J.W.</searchLink><relatesTo>1,2</relatesTo><i> banksj@alum.rpi.edu</i><br /><searchLink fieldCode="AR" term="%22Aslam%2C+T%2E%22">Aslam, T.</searchLink><relatesTo>3</relatesTo><br /><searchLink fieldCode="AR" term="%22Rider%2C+W%2EJ%2E%22">Rider, W.J.</searchLink><relatesTo>2</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Computational+Physics%22">Journal of Computational Physics</searchLink>. Jul2008, Vol. 227 Issue 14, p6985-7002. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Shock+waves%22">Shock waves</searchLink><br /><searchLink fieldCode="DE" term="%22Mechanical+shock%22">Mechanical shock</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+equations%22">Differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Fluid+dynamics%22">Fluid dynamics</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Abstract: A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of Δt 1/(p+1) which yields an estimate for the convergence rate as p/(p +1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations. [Copyright &y& Elsevier] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Computational Physics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jcp.2008.04.002 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 6985 Subjects: – SubjectFull: Shock waves Type: general – SubjectFull: Mechanical shock Type: general – SubjectFull: Differential equations Type: general – SubjectFull: Fluid dynamics Type: general Titles: – TitleFull: On sub-linear convergence for linearly degenerate waves in capturing schemes Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Banks, J.W. – PersonEntity: Name: NameFull: Aslam, T. – PersonEntity: Name: NameFull: Rider, W.J. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 07 Text: Jul2008 Type: published Y: 2008 Identifiers: – Type: issn-print Value: 00219991 Numbering: – Type: volume Value: 227 – Type: issue Value: 14 Titles: – TitleFull: Journal of Computational Physics Type: main |
| ResultId | 1 |
