We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus introduced in Bressan and Marson [Rend. Sem. Mat. Univ. Padova, 94:79-94, 1995]. Numerical results are presented for the one-dimensional Burgers equation and the Euler equations. Using the essential routines of a state-of-the-art code for computational fluid dynamics (CFD) as a starting point, three modifications are required to apply the introduced calculus. First, the CFD code is modified to solve an additional equation for the shock location. Second, we customize the computation of the corresponding tangent to the shock location. Finally, the modified method is enhanced by algorithmic differentiation. Applying the introduced calculus to problems of the Burgers equation and the Euler equations, it is found that correct sensitivities can be computed, whereas the application of black-box algorithmic differentiation fails.

中文翻译：

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双曲流问题的算法微分

我们对开发算法微分框架感兴趣，该框架用于计算双曲偏微分方程组的标量和系统的切向量的近似值。这种数值方法的主要困难是冲击波的存在，通过提出 Bressan 和 Marson [Rend. 准。垫。大学 Padova, 94:79-94, 1995]。给出了一维 Burgers 方程和 Euler 方程的数值结果。使用最先进的计算流体动力学 (CFD) 代码的基本例程作为起点，需要进行三处修改才能应用引入的微积分。首先，修改 CFD 代码以求解冲击位置的附加方程。第二，我们自定义了冲击位置相应切线的计算。最后，修改后的方法通过算法微分得到增强。将引入的微积分应用于Burgers方程和Euler方程的问题，发现可以计算出正确的灵敏度，而黑盒算法微分的应用则失败。