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

Michael Herty, Yizhou Zhou

TL;DR

This work develops a simple numerical method to compute the generalized tangent vector (GTV) for one-dimensional hyperbolic conservation laws, addressing the challenge of interface conditions across shocks. By recasting the GTV evolution with a conservative formulation and explicit, algebraic interface relations, the method preserves jump conditions automatically while advancing the standard solution and the tangent components. The approach is demonstrated on Burgers’ equation and a $2\times2$ system, showing accurate tracking of shock shifts, reliable computation of the tangent vector $\xi$, and $L^1$-convergence of the first-order variation with rate $O(\epsilon^2)$. The results suggest a practical, broadly applicable tool for sensitivity analysis and control in hyperbolic systems, compatible with generic conservative schemes. Future work aims at integrating with higher-order methods to further reduce numerical viscosity and extend applicability to more complex models.

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.

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

TL;DR

This work develops a simple numerical method to compute the generalized tangent vector (GTV) for one-dimensional hyperbolic conservation laws, addressing the challenge of interface conditions across shocks. By recasting the GTV evolution with a conservative formulation and explicit, algebraic interface relations, the method preserves jump conditions automatically while advancing the standard solution and the tangent components. The approach is demonstrated on Burgers’ equation and a system, showing accurate tracking of shock shifts, reliable computation of the tangent vector , and -convergence of the first-order variation with rate . The results suggest a practical, broadly applicable tool for sensitivity analysis and control in hyperbolic systems, compatible with generic conservative schemes. Future work aims at integrating with higher-order methods to further reduce numerical viscosity and extend applicability to more complex models.

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.

Paper Structure

This paper contains 12 sections, 3 theorems, 90 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Interface conditions for the GTV
  4. Numerical method
  5. Single shock case (N=1)
  6. The case of multiple discontinuities
  7. Numerical experiments
  8. Scalar case: Burgers' equation
  9. The $2\times 2$ system with single shock
  10. The $2\times 2$ system with two shocks
  11. Outlook
  12. Details of the proof in Section 3

Key Result

Theorem 2.2

Let $u=u(t,x)$ be a piecewise Lipschitz continuous solution of original with $N$ simple discontinuities. Let $(\bar{v},\bar{\xi})\in L^1(\mathbb{R})\times \mathbb{R}^N$ be a tangent vector to the initial data $u(0,x)=\bar{u}(x)$, generated by the R.V. $\epsilon \mapsto \bar{u}^{\epsilon}$, and call outside the discontinuities of $u$, while, for $\alpha = 1,...,N$, along each line $x = x_{\alpha}

Figures (7)

  • Figure 1: Burger's equation: (left) the exact solution $u^{\epsilon}$ and the first-order variation $u_{\epsilon}$ computed from the GTV at $t=0.2$; (right) the difference $\|u^{\epsilon}-u_{\epsilon}\|_{L^1}$ between the exact solution and the first-order variation with different $\epsilon$.
  • Figure 2: Single shock case for the system \ref{['p-system']}: (left) the numerical solution of $w=(w_\rho,w_q)$ at $t=0.5$ by using Algorithm \ref{['algo1']}. (right) the numerical solution of $v=(v_\rho,v_q)$ at $t=0.05$ by directly computing \ref{['theorem2.1:eq1']}. This test shows the necessity to prescribe the correct interface condition.
  • Figure 3: Single shock case for the system \ref{['p-system']}: the variation $(\rho^{\epsilon}-\rho^0,q^{\epsilon}-q^0)$ by solving the original problem and $(\rho_{\epsilon}-\rho^0,q_{\epsilon}-q^0)$ by computing the generalized tangent vector.
  • Figure 4: Single shock case for the system \ref{['p-system']}: the difference $\|(\rho^{\epsilon}-\rho_{\epsilon},q^{\epsilon}-q_{\epsilon})\|_{L^1}$ between the exact solution and the first-order variation.
  • Figure 5: Two shocks case for the system \ref{['p-system']}: The solution $(\rho^{\epsilon},q^{\epsilon})$ to the system \ref{['p-system']} and the first-order variation $(\rho_{\epsilon},q_{\epsilon})$ computed from the GTV at $t=0.2$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Proof 1
  • Remark 4.1
  • Remark 4.2
  • Proof 2