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

TL;DR

This paper introduces a numerical method to compute the evolution of generalized tangent vectors in one-dimensional hyperbolic PDEs, effectively handling shock waves by combining conservative schemes with new interface conditions.

Contribution

It develops a novel numerical approach for tangent vector computation in hyperbolic systems, addressing challenges posed by shock discontinuities.

Findings

Successfully applied to Burgers' equation

Extended to a 2x2 hyperbolic system with nonlinear fields

Demonstrates accurate tangent vector evolution in numerical tests

Abstract

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.