自由边界流体层的守恒定律
Conservation laws for free-boundary fluid layers
AI总结：
本文将带源项的薄层流体建模为含自由边界的变分不等式，证明其适定性，并揭示消融边缘阻碍精确守恒，进而提出有限体积与有限元格式的后验守恒误差界。
摘要：
受带符号源项影响的薄层流体运动的时变模型，是气候动力学中重要的子问题。例子包括冰盖、海冰，甚至浅水海洋和湖泊。我们将这些问题处理为连续空间弱公式化的离散时间序列，即（单调）变分不等式或互补问题，其中守恒量是层厚。在这类模型中，通常会出现自由边界，在流体层边缘处厚度和质量通量均变为零。在证明这些问题在若干情况下是适定的之后，我们考察了数值格式中离散守恒的局限性。负源区域中的自由边界——即由消融导致的边缘——被证明是连续空间或离散空间意义下精确守恒的障碍。随后，我们提出了可计算的后验量，从而能够给出有限体积和有限元格式中的守恒误差界。
英文摘要：
Time-dependent models of fluid motion in thin layers, subject to signed source terms, represent important sub-problems within climate dynamics. Examples include ice sheets, sea ice, and even shallow oceans and lakes. We address these problems as discrete-time sequences of continuous-space weak formulations, namely (monotone) variational inequalities or complementarity problems, in which the conserved quantity is the layer thickness. Free boundaries wherein the thickness and mass flux both go to zero at the margin of the fluid layer generically arise in such models. After showing these problems are well-posed in several cases, we consider the limitations to discrete conservation in numerical schemes. A free boundary in a region of negative source -- an ablation-caused margin -- turns out to be a barrier to exact conservation in either a continuous- or discrete-space sense. We then propose computable a posteriori quantities which allow conservation-error bounds in finite volume and finite element schemes.
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