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Numerical experiments to "Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit"

Dörich, Benjamin ORCID iD icon 1
1 Institut für Angewandte und Numerische Mathematik (IANM), Karlsruher Institut für Technologie (KIT)

Abstract:

The Kuznetsov equation is a classical wave model of acoustics that incorpo- rates quadratic gradient nonlinearities. When its strong damping vanishes, it undergoes a singular behavior change, switching from a parabolic-like to a hyperbolic quasilinear evolution. In this work, we establish for the first time the optimal error bounds for its finite element approximation as well as a semi-implicit fully discrete approximation that are robust with respect to the vanishing damping parameter. The core of the new arguments lies in devising energy estimates directly for the error equation where one can more easily exploit the polynomial structure of the nonlinearities and compensate inverse estimates with smallness conditions on the error. Numerical experiments are included to illustrate the theoretical results.


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Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Publikationstyp Forschungsdaten
Publikationsdatum 11.01.2024
Erstellungsdatum 08.01.2024
Identifikator DOI: 10.35097/1871
KITopen-ID: 1000167098
Lizenz Creative Commons Namensnennung – Nicht kommerziell – Weitergabe unter gleichen Bedingungen 4.0 International
Projektinformation SFB 1173/3 (DFG, DFG KOORD, SFB 1173/3)
Externe Relationen Siehe auch
Schlagwörter asymptotic-preserving error estimates, full discretization, Kuznetsov equation, nonlinear acoustics
Liesmich

This program is intended to reproduce the results from the preprint

"Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit"
by Benjamin Dörich and Vanja Nikoli\'c

The codes generates the lines in Figures 2, 3, 4, and 5

Requirements

The program is tested with

Kubuntu 22.04.5 and Python 3.10.12 and the following version of its modules:

  • numpy - 1.21.5
  • matplotlib - 3.5.1
  • tikzplotlib - 0.10.1
  • dolfinx - 0.7.2
  • ufl - 2023.2.0
  • mpi4py - 3.1.3

Figures

In the folder "codes_kuznetsov" open a console and run the following commands in seperate consoles.

1) The following can be excuted in paralell:
Run "python3 c1_k1_wave_space_time.py"
Run "python3 c1_k2_wave_space_time.py"
Run "python3 c1_k3_wave_space_time.py"
Run "python3 c1_time_k2_wave_space_time.py"
Run "python3 c3_gauss_ref_comp.py"
Run "python3 c4_wave_beta_conv.py"

2) When the c3_gauss_ref_comp.py has terminated, execute in paralell:
Run "python3 c3_gauss_space.py"
Run "python3 c3_gauss_time.py"

3) Finally, to generate the plots execute:
Run "python3 c99_generate_plots.py"
Run "python3 c99_generate_gauss_plots.py"

After running the calculations, the files are stored as tikz and pdf files in the folder "tikz/".
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