| Issue |
ESAIM: M2AN
Volume 59, Number 1, January-February 2025
|
|
|---|---|---|
| Page(s) | 579 - 612 | |
| DOI | https://doi.org/10.1051/m2an/2024083 | |
| Published online | 16 January 2025 | |
A numerical view on α-dissipative solutions of the Hunter–Saxton equation
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
* Corresponding author: katrin.grunert@ntnu.no
Received:
1
August
2024
Accepted:
3
December
2024
We propose a new numerical method for α-dissipative solutions of the Hunter–Saxton equation, where α belongs to W1,∞(R, [0, 1)). The method combines a projection operator with a generalized method of characteristics and an iteration scheme, which is based on enforcing minimal time steps whenever breaking times cluster. Numerical examples illustrate that these minimal time steps increase the efficiency of the algorithm substantially. Moreover, convergence of the wave profile is shown in C([0, T], L∞(R)) for any finite T ≥ 0.
Mathematics Subject Classification: 65M12 / 65M25 / 65M06 / 35Q35
Key words: Hunter–Saxton equation / α-dissipative solutions / numerical method / convergence analysis / minimal time steps
© The authors. Published by EDP Sciences, SMAI 2025
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