A Convergent Numerical Algorithm for α-Dissipative Solutions of the Hunter–Saxton Equation

A convergent numerical method for α -dissipative solutions of the Hunter–Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step t...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 99; no. 1; p. 14
Main Authors: Christiansen, Thomas, Grunert, Katrin, Nordli, Anders, Solem, Susanne
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2024
Springer Nature B.V
Subjects:
Algorithms
Approximation
Cauchy problems
Computational Mathematics and Numerical Analysis
Convergence
Energy
Energy dissipation
Inertia
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Method of characteristics
Numerical analysis
Numerical methods
Theoretical
Primary: 65M12
Projection operator
65M25
35Q35
Secondary: 65M06
Conservative solutions
Numerical method
Dissipative solutions
Hunter–Saxton equation
Convergence
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:A convergent numerical method for α -dissipative solutions of the Hunter–Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that errors due to the energy dissipation at later times remain small. Furthermore, it is shown that the main quantity of interest, the wave profile, converges in L ∞ for all t ≥ 0 , while a subsequence of the energy density converges weakly for almost every time.
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content type line 14
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02479-4