This work is concerned with the computation of the first-order variation for one-dimensional hyperbolic partial differential equations. In the case of shock waves the main challenge is addressed by developing a numerical method to compute the evolution of the generalized tangent vector introduced by Bressan and Marson (1995). Our basic strategy is to combine the conservative numerical schemes and a novel expression of the interface conditions for the tangent vectors along the discontinuity. Based on this, we propose a simple numerical method to compute the tangent vectors for general hyperbolic systems. Numerical results are presented for Burgers' equation and a 2 x 2 hyperbolic system with two genuinely nonlinear fields.

翻译：本研究关注一维双曲型偏微分方程一阶变分的计算问题。针对激波情形，通过建立数值方法计算Bressan与Marson（1995）提出的广义切向量演化过程，解决了该问题的核心难点。我们的基本策略是结合守恒型数值格式与间断处切向量界面条件的新颖表达形式。在此基础上，针对一般双曲系统提出了一种简洁的切向量数值计算方法。文中以Burgers方程和具有两个真非线性场的2×2双曲系统为例展示了数值计算结果。