Numerical methods for conservation laws with rough flux

H. Hoel; K. H. Karlsen; N. H. Risebro; E. B. Storrøsten

doi:10.1007/s40072-019-00145-7

Numerical methods for conservation laws with rough flux

H. Hoel, K. H. Karlsen, N. H. Risebro, E. B. Storrøsten

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for α-Hölder continuous paths the convergence rate of the numerical methods can improve from O(COST -γ) , for some γ∈ [α/ (12 - 8 α) , α/ (10 - 6 α)] , with α∈ (0 , 1) , to O(COST -min(1/4,α/2)). Numerical examples support the theoretical results.

Original language	English (US)
Pages (from-to)	186-261
Number of pages	76
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	8
Issue number	1
DOIs	https://doi.org/10.1007/s40072-019-00145-7
State	Published - Jun 14 2019
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): CRG4 Award Ref: 2584
Acknowledgements: This work received supported by the Research Council of Norway through the project Stochastic Conservation Laws (250674/F20) and by the KAUST CRG4 Award Ref: 2584.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1007/s40072-019-00145-7

https://repository.kaust.edu.sa/bitstream/10754/630807/1/1802.00708.pdf

Cite this

@article{f6d06693d8d0456892be9262003cf134,

title = "Numerical methods for conservation laws with rough flux",

abstract = "Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for α-H{\"o}lder continuous paths the convergence rate of the numerical methods can improve from O(COST -γ) , for some γ∈ [α/ (12 - 8 α) , α/ (10 - 6 α)] , with α∈ (0 , 1) , to O(COST -min(1/4,α/2)). Numerical examples support the theoretical results.",

author = "H. Hoel and Karlsen, {K. H.} and Risebro, {N. H.} and Storr{\o}sten, {E. B.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): CRG4 Award Ref: 2584 Acknowledgements: This work received supported by the Research Council of Norway through the project Stochastic Conservation Laws (250674/F20) and by the KAUST CRG4 Award Ref: 2584. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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doi = "10.1007/s40072-019-00145-7",

language = "English (US)",

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AU - Hoel, H.

AU - Karlsen, K. H.

AU - Risebro, N. H.

AU - Storrøsten, E. B.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): CRG4 Award Ref: 2584 Acknowledgements: This work received supported by the Research Council of Norway through the project Stochastic Conservation Laws (250674/F20) and by the KAUST CRG4 Award Ref: 2584. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2019/6/14

Y1 - 2019/6/14

N2 - Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for α-Hölder continuous paths the convergence rate of the numerical methods can improve from O(COST -γ) , for some γ∈ [α/ (12 - 8 α) , α/ (10 - 6 α)] , with α∈ (0 , 1) , to O(COST -min(1/4,α/2)). Numerical examples support the theoretical results.

AB - Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for α-Hölder continuous paths the convergence rate of the numerical methods can improve from O(COST -γ) , for some γ∈ [α/ (12 - 8 α) , α/ (10 - 6 α)] , with α∈ (0 , 1) , to O(COST -min(1/4,α/2)). Numerical examples support the theoretical results.

UR - http://hdl.handle.net/10754/630807

UR - http://link.springer.com/10.1007/s40072-019-00145-7

UR - http://www.scopus.com/inward/record.url?scp=85068166415&partnerID=8YFLogxK

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DO - 10.1007/s40072-019-00145-7

M3 - Article

SN - 2194-0401

VL - 8

SP - 186

EP - 261

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

IS - 1

ER -

Numerical methods for conservation laws with rough flux

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