截断Westervelt系统的Rothe时间离散化与弱解
Rothe time discretization and weak solutions for a cutoff Westervelt system
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中文总结 AI 辅助
该研究针对截断Westervelt系统提出全隐式Rothe时间离散化方法，证明其适定性与弱解存在性，给出弱-强唯一性准则，通过数值实验验证格式的稳定性与时间行为。
中文摘要 AI 辅助
我们研究了Westervelt方程的截断一阶形式的全隐式Rothe时间离散化方法。该方法的关键要素是焓变量和原始迁移率变量，它们将每个非线性时间步转化为一致单调的椭圆问题，避免了高阶能量估计和逆不等式。对于每个时间步，离散问题可简化为单调椭圆方程，借助Browder-Minty定理可得到Rothe格式的适定性。我们推导了离散能量不等式，通过Alt-Luckhaus型论证建立了变换后变量的紧性，并取极限得到截断一阶系统弱解的存在性。我们证明了截断问题的弱-强唯一性原理，并提出了一个条件后验准则，在此准则下截断不起作用。对于受迫截断问题的充分正则解，我们建立了一致性估计，该估计可识别Rothe近似在自然的离散时间弱范数中产生的时间残差，为观测到的时间行为提供了严格的一致性基础。最后，数值实验验证了该格式的稳定性、观测到的近一阶时间行为，以及在测试 regime 中计算得到的离散轨迹上截断不起作用的特性。
英文摘要
We study a fully implicit Rothe time discretization for a cutoff first-order formulation of the Westervelt equation. The key ingredients are the enthalpy variable and the primitive mobility variable, which turn each nonlinear time step into a uniformly monotone elliptic problem and avoid higher-order energy estimates and inverse inequalities. For every time step, the discrete problem reduces to a monotone elliptic equation, which yields well-posedness of the Rothe scheme by the Browder-Minty theorem. We derive a discrete energy inequality, establish compactness for the transformed variable by an Alt-Luckhaus type argument, and pass to the limit to obtain existence of weak solutions for the cutoff first-order system. We prove a weak-strong uniqueness principle for the cutoff problem and formulate a conditional a posteriori criterion under which the cutoff is inactive. For sufficiently regular solutions of the forced cutoff problem, we establish consistency estimates which identify the temporal residuals produced by the Rothe approximation in the natural discrete-in-time weak norms. These estimates provide a rigorous consistency basis for the observed temporal behavior. Finally, numerical experiments illustrate the stability of the scheme, its observed near first-order temporal behavior, and inactivity of the cutoff along the computed discrete trajectories in the tested regimes.
