关于非均匀波动方程的未映射帐篷倾斜算法
On the unmapped tent pitching for the heterogeneous wave equation
AI总结：
提出未映射帐篷倾斜（UTP）算法在非均匀介质中的扩展策略，通过使用最大传播速度确定统一时空子域尺寸，实现高效并行求解。
摘要：
未映射帐篷倾斜（UTP）算法是一种用于双曲问题并行求解的时空域分解方法。它最初是在[Ciaramella, Gander, Mazzieri, 2024]中针对齐次一维波动方程引入的。UTP受映射帐篷倾斜（MTP）算法[Gopalakrishnan, Schöberl, Wintersteiger, 2017]启发，后者通过迭代构建多面体时空子域（称为帐篷）来构造解。在MTP中，每个物理帐篷被映射到一个时空矩形上，在那里求解局部问题，然后再映射回原始域。相比之下，UTP避免了非线性且可能奇异的映射步骤，通过直接在包含帐篷的物理时空矩形上计算解，代价是在帐篷外部区域进行冗余计算。在本工作中，我们研究了将UTP扩展到非均匀介质的几种策略，其中波传播速度在域的两个子区域上是分段常数。在所考虑的方法中，就计算时间而言最有效的是在两个区域中使用由最大传播速度确定的相同空间和时间维度的时空子域的方法。
英文摘要：
The Unmapped Tent Pitching (UTP) algorithm is a space--time domain decomposition method for the parallel solution of hyperbolic problems. It was originally introduced for the homogeneous one-dimensional wave equation in [Ciaramella, Gander, Mazzieri, 2024]. UTP is inspired by the Mapped Tent Pitching (MTP) algorithm [Gopalakrishnan, Sch{ö}berl, Wintersteiger, 2017], which constructs the solution by iteratively building polytopal space--time subdomains, referred to as tents. In MTP, each physical tent is mapped onto a space--time rectangle, where local problems are solved before being mapped back to the original domain. In contrast, UTP avoids the nonlinear and potentially singular mapping step by computing the solution directly on a physical space--time rectangle that contains the tent, at the expense of redundant computations in the region outside the tent. In this work, we investigate several strategies to extend UTP to heterogeneous media, where the wave propagation speed is piecewise constant over two subregions of the domain. Among the considered approaches, the most efficient in terms of computational time is the one employing space--time subdomains with identical spatial and temporal dimensions in both regions, determined by the maximum propagation speed.
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