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

Communications in Mathematical Sciences, vol. 24, pp. 1923–1943

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 [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.

Authors 2

1. Michaël Herty Aachen Chair in Numerical Analysis Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University · University of Pretoria

   Affiliation as printed

   Chair in Numerical Analysis, Institut für Geometrie und Praktische Mathematik (IGPM), RWTH Aachen University, Im Süsterfeld 2, D-52072 Aachen, Germany; Extraordinary Professor, Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield 0028, South Africa

2. Yizhou Zhou Aachen

   RWTH Aachen University

   Affiliation as printed

   IGPM, RWTH Aachen University, Im Süsterfeld 2, D-52072 Aachen, Germany