Abstract
In this work, we use exact solutions of one-dimensional Burgers equation to train an artificial neuron as a shock wave detector. The expression of the artificial neuron detector is then modified into a practical form to reflect admissible jump of eigenvalues. We show the working mechanism of the practical form is consistent with compressing or intersecting of characteristic curves. In addition, we prove there is indeed a discontinuity inside the cell detected by the practical form, and smooth extrema and large gradient regions are never marked. As a result, we apply the practical form to numerical schemes as a shock wave indicator with its easy extension to multi-dimensional conservation laws. Numerical results are present to demonstrate the robustness of the present indicator under Runge–Kutta Discontinuous Galerkin framework, its performance is generally compared to TVB-based indicators more efficiently and accurately. To treat the initial inadmissible jumps, including linear contact discontinuities and those evolving into rarefaction waves, a preliminary strategy of combining a traditional indicator in the beginning with the present indicator is suggested. We believe the present indicator can be applied to unstructured mesh in the future.
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Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M.: TensorFlow, Large-scale machine learning on heterogeneous systems, (2015) https://www.tensorflow.org/
Chang, T., Hsiao, L.: The Riemann problem and interaction of waves in gas dynamics. NASA STI/Recon Technical Report A 90 (1989)
Cheng, J., Du, Z., Lei, X., Wang, Y., Li, J.: A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible Euler equations. Comput. Fluids 181, 248–258 (2019)
Cockburn, B., Hou, S., Shu, C.W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54, 545–581 (1990)
Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)
Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52, 411–435 (1989)
Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (2010)
Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50, 544–573 (2012)
Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
Gurney, K.: An Introduction to Neural Networks[M]. UCL Press, Boca Raton (1997)
Harten, A.: On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21, 1–23 (1984)
Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes. I. In: Upwind and High-Resolution Schemes, pp. 187–217. Springer (1997)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)
Jameson, A., Schmidt, W., Turkel, E.: Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes, In: 14th Fluid and Plasma Dynamics Conference, p. 1259 (1981)
Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48, 323–338 (2004)
Lax, P.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math 7, 159–193 (1954)
Leveque, R.J.: Finite volume methods for hyperbolic problems. Meccanica 39, 88–89 (2004)
Nair, V., Hinton, G.E.: Rectified linear units improve restricted boltzmann machines. In: International Conference on International Conference on Machine Learning (2010)
Nasr, G.E., Badr, E.A., Joun, C.: Cross entropy error function in neural networks: forecasting gasoline demand. In: Fifteenth International Florida Artificial Intelligence Research Society Conference (2002)
Nowlan, S.J., Hinton, G.E.: Simplifying Neural Networks by Soft Weight-Sharing. MIT Press, Cambridge (1992)
Oh, S.H.: Error back-propagation algorithm for classification of imbalanced data. Neurocomputing 74, 1058–1061 (2011)
Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biol. 52, 99–115 (1990)
Qiu, J., Shu, C.-W.: Hermite weno schemes and their application as limiters for Runge–Kutta discontinuous galerkin method: one-dimensional case. J. Comput. Phys. 193, 115–135 (2004)
Qiu, J., Shu, C.-W.: A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput. 27, 995–1013 (2005)
Qiu, J., Shu, C.-W.: Hermite weno schemes and their application as limiters for Runge–Kutta discontinuous galerkin method ii: two dimensional case. Comput. Fluids 34, 642–663 (2005)
Qiu, J., Shu, C.-W.: Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26, 907–929 (2005)
Ray, D., Hesthaven, J.S.: An artificial neural network as a troubled-cell indicator. J. Comput. Phys. 367, 166–191 (2018)
Ray, D., Hesthaven, J.S.: Detecting troubled-cells on two-dimensional unstructured grids using a neural network. J. Comput. Phys. 397, 108845 (2019)
Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)
Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325–432. Springer (1998)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, ii. In: Upwind and High-Resolution Schemes, pp. 328–374. Springer (1989)
Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)
Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)
Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, pp. 87–114. Springer, Berlin (2013)
Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)
Woodward, C.P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)
Acknowledgements
The authors would like to acknowledge the reviewers for the valuable comments and suggestions. This work is supported by National Numerical Wind Tunnel Project, National Nature Science Foundation of China (Nos. U1730118 and 91530325), Fundamental Research of Civil Aircraft (No. MJ-F-2012-04).
Funding
This work was funded by National Numerical Wind Tunnel Project, National Nature Science Foundation of China (Nos. U1730118 and 91530325), Fundamental Research of Civil Aircraft (No. MJ-F-2012-04).
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Submitted to the editors on May 9 2019.
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Feng, Y., Liu, T. & Wang, K. A Characteristic-Featured Shock Wave Indicator for Conservation Laws Based on Training an Artificial Neuron. J Sci Comput 83, 21 (2020). https://doi.org/10.1007/s10915-020-01200-5
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DOI: https://doi.org/10.1007/s10915-020-01200-5
Keywords
- Hyperbolic conservation laws
- Characteristic curve
- Discontinuous Galerkin
- Troubled-cell indicator
- Artificial neural networks