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Updated 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)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsdaten
Publikationsdatum 16.09.2026
Erstellungsdatum 16.09.2026
Identifikator DOI: 10.35097/5cdgzm5cjb87yaf0
KITopen-ID: 1000197041
Lizenz Creative Commons Namensnennung – Nicht kommerziell – Weitergabe unter gleichen Bedingungen 4.0 International
Projektinformation SFB 1173, 258734477 (DFG, DFG KOORD, SFB 1173/3)
Externe Relationen Siehe auch
Schlagwörter asymptotic-preserving error estimates, full discretization, Kuznetsov equation, nonlinear acoustics
Liesmich

Readme

This program is intended to reproduce the results from the paper

"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 easiest way to reproduce the results is via a conda enviroment which can be set up by executing

conda create --name kuznetsov -c conda-forge \
python=3.10.12 \
numpy=1.23.5 \
matplotlib=3.5.1 \
tikzplotlib=0.10.1 \
fenics-dolfinx=0.7.2 \
fenics-ufl=2023.2.0 \
mpi4py=3.1.3 \
h5py=3.10.0 \
pyvista=0.42.3 \
webcolors=1.12

conda activate kuznetsov

Figures

In the folder "codes_kuznetsov" open a console and

Version1: 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_space.py"
    Run "python3 c99_generate_plots_time.py"
    Run "python3 c99_generate_gauss_plots.py"

Version 2: Alternatively to 1) - 3), one can also use the bash script "run_all.sh" via

chmod +x run_all.sh
./run_all.sh

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