A Wave Front Tracking Scheme for Flux Reconstruction in $2\times 2$ Hyperbolic Conservation Laws
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915631990833152 |
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| author | Duan, Chaohua Jiang, Yan Liu, Hongyu Peng, Wenjian |
| author_facet | Duan, Chaohua Jiang, Yan Liu, Hongyu Peng, Wenjian |
| contents | This paper introduces a novel wave front tracking framework for reconstructing unknown flux functions in $2\times 2$ hyperbolic conservation laws, extending beyond the well-studied scalar case. By analyzing Riemann solutions at fixed observation times, we develop explicit reconstruction formulas that handle arbitrary combinations of shock and rarefaction waves through a unified equivalent shock concept. Our method constructs piecewise quadratic $C^1$ flux approximations with rigorous convergence guarantees: the approximation errors decrease quadratically with the discretization parameters for function values and linearly for derivatives under $C^{1,1}$ regularity, with enhanced cubic and quadratic convergence respectively under $C^3$ regularity. Applications to the isentropic Euler equations and the mathematically equivalent p-system in compressible fluid dynamics demonstrate the method's capability to identify complete equations of state from limited dynamic measurements, providing a systematic approach to a fundamental inverse problem in continuum mechanics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Wave Front Tracking Scheme for Flux Reconstruction in $2\times 2$ Hyperbolic Conservation Laws Duan, Chaohua Jiang, Yan Liu, Hongyu Peng, Wenjian Analysis of PDEs Numerical Analysis This paper introduces a novel wave front tracking framework for reconstructing unknown flux functions in $2\times 2$ hyperbolic conservation laws, extending beyond the well-studied scalar case. By analyzing Riemann solutions at fixed observation times, we develop explicit reconstruction formulas that handle arbitrary combinations of shock and rarefaction waves through a unified equivalent shock concept. Our method constructs piecewise quadratic $C^1$ flux approximations with rigorous convergence guarantees: the approximation errors decrease quadratically with the discretization parameters for function values and linearly for derivatives under $C^{1,1}$ regularity, with enhanced cubic and quadratic convergence respectively under $C^3$ regularity. Applications to the isentropic Euler equations and the mathematically equivalent p-system in compressible fluid dynamics demonstrate the method's capability to identify complete equations of state from limited dynamic measurements, providing a systematic approach to a fundamental inverse problem in continuum mechanics. |
| title | A Wave Front Tracking Scheme for Flux Reconstruction in $2\times 2$ Hyperbolic Conservation Laws |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2511.15261 |