Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations
Abstract
:1. Introduction
2. Direct Problem
2.1. Problem Formulation
2.2. Methods for Solving the Direct Problem
- Obtaining the flux values by solving the Riemann problem;
- Updating the state variables on the next time step, using the solution of the Riemann problem.
- Reconstruction of the state variables on the current time step, using piecewise-linear extrapolation;
- Evolution of the reconstructed state variables by conservation laws with a time ;
- Obtaining the flux values for the “midpoint” time step by solving the Riemann problem for evolved state variables;
- Updating the state values from the current time step on the next time step by conservation laws with a time .
3. Inverse Problems
4. Numerical Results
5. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Mesh Dependence-Exact Data | |||
---|---|---|---|
Mesh Size | Residual | Relative Error (Density) | Relative Error (Speed of Sound) |
−4.95 | 0.056 | 0.049 | |
−4.88 | 0.045 | 0.039 | |
−4.85 | 0.034 | 0.029 |
Mesh Dependence-5% Noise in the Data | |||
---|---|---|---|
Mesh Size | Residual | Relative Error (Density) | Relative Error (Speed of Sound) |
−3.51 | 0.058 | 0.052 | |
−3.16 | 0.047 | 0.041 | |
−2.86 | 0.036 | 0.031 |
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Klyuchinskiy, D.; Novikov, N.; Shishlenin, M. Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations. Mathematics 2021, 9, 199. https://doi.org/10.3390/math9020199
Klyuchinskiy D, Novikov N, Shishlenin M. Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations. Mathematics. 2021; 9(2):199. https://doi.org/10.3390/math9020199
Chicago/Turabian StyleKlyuchinskiy, Dmitriy, Nikita Novikov, and Maxim Shishlenin. 2021. "Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations" Mathematics 9, no. 2: 199. https://doi.org/10.3390/math9020199
APA StyleKlyuchinskiy, D., Novikov, N., & Shishlenin, M. (2021). Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations. Mathematics, 9(2), 199. https://doi.org/10.3390/math9020199