Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit

TL;DR

This work derives $\beta$-robust finite element and semi-implicit fully discrete error bounds for the Kuznetsov equation in the inviscid limit, ensuring stable numerical performance as damping vanishes. The authors introduce energy estimates directly for the error equation, leveraging the polynomial structure of the nonlinearities and inverse-estimate control via small error, to obtain optimal $h^{2k}$ convergence with $k\ge 2$ that are uniform in $\beta$. They also prove asymptotic-preserving behavior: as $\beta\to 0$, the discrete solutions converge linearly in $\beta$ to the inviscid counterparts, both in semi-discrete and fully discrete settings, under appropriate CFL-type time-c-step relations. Numerical experiments with quadratic/ higher-order elements validate the theoretical rates and the uniform-in-$\beta$ performance, including inviscid-limit convergence and large-domain Gaussian-pulse simulations, underscoring the practical impact for nonlinear acoustic models in low-dissipation regimes.

Abstract

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.

Figures (5)

• Figure 1: Diagram representing the main contributions of this work
• Figure 2: Convergence of \ref{['eq:Kuznetsov_space_discr_full_eq']} with $\| \nabla \partial_t u(t) - \nabla \partial_t u_h (t) \|_{L^2(\Omega)}$ at $t=0.8$ with the parameters $c_{\mathrm{sp}} = 0.1$, $c_{\mathrm{time}} = 0.5$ in \ref{['eq:sol_shock']} for elements of order $k=1,2,3$ and $\tau \approx 1.5 \cdot 10^{-3}$ and damping parameters $\beta = 0 , 10^{-3}, 10^{-2}$ (from left to right). The dashed lines indicate order $\mathcal{O}(h^k)$ for $k=1,2,3$.
• Figure 3: Convergence of \ref{['eq:Euler']} with $\| \nabla \partial_t u(t_{n}) - \nabla \partial_{\tau} u_h^{n} \|_{L^2(\Omega)}$ for $n = N+1$ with the parameters $c_{\mathrm{sp}} = 0.1$, $c_{\mathrm{time}} = 0.5$ in \ref{['eq:sol_shock']} with $k=2$ and $h \approx 1.1 \cdot 10^{-2}$ and damping parameters $\beta = 0 , 10^{-3}, 10^{-2}$ (from left to right). The dashed lines indicate order $\mathcal{O}(\tau)$.
• Figure 4: Convergence of $(u_{h,\beta=0}^{n},\partial_{\tau} u_{h,\beta=0}^{n})$ and $(u_{h,\beta}^{n},\partial_{\tau} u_{h,\beta}^{n})$ in the $H^1(\Omega) \times L^2(\Omega)$-norm at the end time $t=0.8$ for different values of $h$ and $\tau$. The dashed line indicates order $\mathcal{O}(\beta)$.
• Figure 5: Left: Convergence of \ref{['eq:Kuznetsov_space_discr_full_eq']} with $\| \nabla \partial_t u(t) - \nabla \partial_t u_h (t) \|_{L^2(\Omega)}$ at $t=0.8$ for elements of order $k=2$ and $\tau \approx 7.8 \cdot 10^{-4}$ and damping parameters $\beta = 0 , 10^{-3}, 10^{-2}$. The dashed line indicates order $\mathcal{O}(h^2)$. Right: Convergence of \ref{['eq:Euler']} with $\| \nabla \partial_t u(t_{n}) - \nabla \partial_{\tau} u_h^{n} \|_{L^2(\Omega)}$ for $n = N+1$ with $k=2$ and $h \approx 9 \cdot 10^{-2}$ and damping parameters $\beta = 0 , 10^{-3}, 10^{-2}$. The dashed line indicates order $\mathcal{O}(\tau)$.

Theorems & Definitions (48)

• Theorem 2.1: Robust finite element estimates
• Theorem 2.2: Asymptotic-preserving behavior in the inviscid limit
• Theorem 2.3: Robust fully discrete error bounds
• Theorem 2.4: Asymptotic-preserving behavior in the inviscid limit
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof