A numerical method for solving the generalized tangent vector of hyperbolic systems

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 [A. Bressan and A. Marson, Comm. Part. Diff. Eqs., 20(9-10):1491-1552, 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 $2\times2$ hyperbolic system with two genuinely nonlinear fields.