关于寒冷、浅层且热耦合冰盖的精确解与数值计算
On exact solutions and numerics for cold, shallow, and thermocoupled ice sheets
AI总结：
本文构造热耦合浅层冰盖连续介质模型的精确解 Test G，提供 C 参考实现，并分析温度方程有限差分格式的 CFL 条件、误差增长与收敛性。
摘要：
这份分为三节的报告可视为（Bueler、Brown 和 Lingle 2006）的扩展附录。首先，我们详细构造了寒冷、浅层且热耦合冰盖的标准连续介质模型的一个精确解。该构造通过计算补偿性累积函数和热源函数来完成，这些函数使所选取的一对厚度函数和温度函数成为该耦合系统的精确解。我们在此构造的解是（Bueler 等人，2006）中的“Test G”，而稳态解“Test F”是其特例。在第二节中，我们给出了这些精确解的参考 C 语言实现。在最后一节中，我们对热耦合模型中温度方程的有限差分格式进行了误差分析。该误差分析给出三项结果：第一，给出了平流格式稳定性所需的 Courant-Friedrichs-Lewy（CFL）条件的正确形式；第二，给出了一个误差增长方程，有助于理解（Payne 等人，2000）中著名的“辐条”现象；第三，在严格的固定几何形状和平滑性假设下给出了一个收敛定理。
英文摘要：
This three section report can be regarded as an extended appendix to (Bueler, Brown, and Lingle 2006). First we give the detailed construction of an exact solution to a standard continuum model of a cold, shallow, and thermocoupled ice sheet. The construction is by calculation of compensatory accumulation and heat source functions which make a chosen pair of functions for thickness and temperature into exact solutions of the coupled system. The solution we construct here is ``TestG'' in (Bueler and others, 2006) and the steady state solution ``Test F'' is a special case. In the second section we give a reference C implementation of these exact solutions. In the last section we give an error analysis of a finite difference scheme for the temperature equation in the thermocoupled model. The error analysis gives three results, first the correct form of the Courant-Friedrichs-Lewy (CFL) condition for stability of the advection scheme, second an equation for error growth which contributes to understanding the famous ``spokes'' of (Payne and others, 2000), and third a convergence theorem under stringent fixed geometry and smoothness assumptions.
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