A convergence rate result for front tracking approximations of conservation laws with discontinuous flux

Shyam Sundar Ghoshal; John D Towers

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arXiv · 2509.22952

A convergence rate result for front tracking approximations of conservation laws with discontinuous flux

Shyam Sundar Ghoshal·John D Towers

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Abstract

We consider the initial value problem for a scalar conservation law in one space dimension with a single spatial flux discontinuity, the so-called two-flux problem. We prove that a well-known front tracking algorithm has a convergence rate of at least one-half. The fluxes are required to be smooth, but are not required to be convex or concave, monotone, or even unimodal. We require that there are no more than finitely many flux crossings, but we do not require that they satisfy the so-called crossing condition. If both fluxes are strictly increasing or strictly decreasing then our analysis yields a convergence rate of one, in agreement with a recent result. Similarly, if the fluxes are equal, i.e., there is no flux discontinuity, we obtain a convergence rate of one in this case also, in agreement with a classical result. The novelty of this paper is that the class of discontinuous-flux conservation laws for which there is a front tracking error estimate is expanded, and that the method of analysis is new; we do not use the Kuznetsov lemma which is commonly used for this type of analysis.

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Shyam Sundar Ghoshal, John D Towers. 2025-09-26. A convergence rate result for front tracking approximations of conservation laws with discontinuous flux. https://arxiv.org/abs/2509.22952

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