Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912616171962368 |
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| author | Dörich, Benjamin Nikolić, Vanja |
| author_facet | Dörich, Benjamin Nikolić, Vanja |
| contents | The Kuznetsov equation is a classical wave model of acoustics that incorporates 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit Dörich, Benjamin Nikolić, Vanja Numerical Analysis 65M12, 65M15, 35L72, 65M60 The Kuznetsov equation is a classical wave model of acoustics that incorporates 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. |
| title | Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit |
| topic | Numerical Analysis 65M12, 65M15, 35L72, 65M60 |
| url | https://arxiv.org/abs/2401.06492 |