多维双曲平衡律的灵敏度微积分及其数值实现
Sensitivity Calculus and its Numerical Implementation for Multi-D Hyperbolic Balance Laws
- RWTH Aachen University（亚琛工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对多维双曲平衡律，提出基于法向位移的灵敏度微积分方法，并开发数值算法计算初值一阶变分，二维实验验证一阶精度。
AI中文摘要：
我们从解析和数值两方面研究了多维标量平衡律的解对初始数据扰动的灵敏度。可靠的灵敏度一阶信息对于受双曲平衡律约束的基于梯度的优化和反问题至关重要。在双曲问题中，此类扰动既影响解的光滑分量，也影响激波的位置。因此，解算子的经典差商即使在空间一维情况下通常也无法在$L^1$中收敛。尽管针对一维问题已发展了广义切向量技术，但将这些思想扩展到多维空间要困难得多，因为这需要对不连续超曲面进行几何描述。我们用法向位移来表示不连续超曲面的扰动，并考虑由此引起的法向方向的变化。这种表示方法导出了广义切向量各分量的演化方程。基于该微积分，我们开发了一种数值方法，用于计算关于初始数据的一阶变分。二维空间中的数值实验证实了所得近似具有预期的一阶精度。
英文摘要：
We investigate the sensitivity of solutions to multi-dimensional scalar balance laws with respect to perturbations of the initial data analytically and numerically. Reliable first-order sensitivity information is essential in gradient-based optimization and inverse problems constrained by hyperbolic balance laws. In hyperbolic problems, such perturbations affect both the smooth components of the solution and the locations of shocks. Consequently, classical difference quotients of the solution operator generally fail to converge in $L^1$, even in one space dimension. Although generalized tangent-vector techniques have been developed for one-dimensional problems, extending these ideas to multiple space dimensions is considerably more challenging because it requires a geometric description of hypersurfaces of discontinuity. We represent perturbations of hypersurfaces of discontinuity by normal displacements and account for the induced variation of the normal direction. This representation yields evolution equations for the components of a generalized tangent vector. Based on this calculus, we develop a numerical method for computing first-order variations with respect to the initial data. Numerical experiments in two space dimensions confirm the expected first-order accuracy of the resulting approximation.
