Abstract
This paper develops and analyzes a new numerical scheme for solving hyperbolic conservation laws that combines the Lax Wendroff method with \(l_1\) regularization. While prior investigations constructed similar algorithms, the method developed here adds a new critical conservation constraint. We demonstrate that the resulting method is equivalent to the well known lasso problem, guaranteeing both existence and uniqueness of the numerical solution. We further prove consistency, convergence, and conservation of our scheme, and also show that it is TVD and satisfies the weak entropy condition for conservation laws. Numerical solutions to Burgers’ and Euler’s equation validate our analytical results.
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Notes
For simplicity we consider only odd orders which yield symmetric stencil for uniformly distributed grid points.
References
Archibald, R., Gelb, A., Platte, R.B.: Image reconstruction from undersampled Fourier data using the polynomial annihilation transform. J. Sci. Comput. 67(2), 432–452 (2016)
Archibald, R., Gelb, A., Yoon, J.: Polynomial fitting for edge detection in irregularly sampled signals and images. SIAM J. Numer. Anal. 43(1), 259–279 (2005)
Arvanitis, C., Makridakis, C., Sfakianakis, N.I.: Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws. J. Hyperb. Differ. Equ. 07(03), 383–404 (2010)
Candes, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted $\ell _1$ minimization. J. Fourier Anal. Appl. 14(5), 877–905 (2008)
Don, W.-S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate Fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)
Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33(1), 1–22 (2010)
Gottlieb, D., Shu, C.-W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)
Guermond, J.-L., Marpeau, F., Popov, B.: A fast algorithm for solving first-order PDEs by l1-minimization. Commun. Math. Sci. 6(1), 199–216 (2008)
Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time-Dependent Problems and Difference. Wiley, London (2013)
Hou, T.Y., Li, Q., Schaeffer, H.: Sparse + low-energy decomposition for viscous conservation laws. J. Comput. Phys. 288(C), 150–166 (2015)
Lavery, J.E.: Solution of steady-state one-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 26(5), 1081–1089 (1989)
Lavery, J.E.: Solution of steady-state, two-dimensional conservation laws by mathematical programming. SIAM J. Numer. Anal. 28(1), 141–155 (1991)
LeVeque, R.J.: Numerical Methods for Conservation Laws. Springer, Berlin (1992)
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Society for Industrial and Applied Mathematics, Philadelphia (2007)
Liska, R., Wendroff, B.: Composite schemes for conservation laws. SIAM J. Numer. Anal. 35(6), 2250–2271 (1998)
Scarnati, T., Gelb, A., Platte, R.B.: Using $l1$ regularization to improve numerical partial differential equation solvers. J. Sci. Comput. 75, 225–252 (2018)
Schaeffer, H., Caflisch, R., Hauck, C.D., Osher, S.: Sparse dynamics for partial differential equations. Proc. Nat. Acad. Sci. 110(17), 6634–6639 (2013)
Shu, C.-W., Wong, P.S.: A note on the accuracy of spectral method applied to nonlinear conservation laws. J. Sci. Comput. 10(3), 357–369 (1995)
Tadmor, E.: Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26(1), 30–44 (1989)
Tadmor, E.: Shock capturing by the spectral viscosity method. Comput. Methods Appl. Mech. Eng. 80(1), 197–208 (1990)
Tadmor, E., Waagan, K.: Adaptive spectral viscosity for hyperbolic conservation laws. SIAM J. Sci. Comput. 34(2), A993–A1009 (2012)
Tibshirani, R.J.: The lasso problem and uniqueness. Electron. J. Stat. 7, 1456–1490 (2013)
van den Berg, E., Friedlander, M. P.: SPGL1: a solver for large-scale sparse reconstruction (2007, June). http://www.cs.ubc.ca/labs/scl/spgl1
van den Berg, E., Friedlander, M.P.: Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)
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In memory of Saul Abarbanel, a great scholar, mentor, and friend.
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The work of X. H. and Q. L. is supported in part by NSF-DMS 1619778, KI-Net RNMS-1107291 and NSF-TRIPODS 1740707. The work of A.G. is supported in part by National Science Foundation under the Grant NSF-DMS 1502640, NSF-DMS 1732434, and AFOSR FA9550-18-1-0316.
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Gelb, A., Hou, X. & Li, Q. Numerical Analysis for Conservation Laws Using \(l_1\) Minimization. J Sci Comput 81, 1240–1265 (2019). https://doi.org/10.1007/s10915-019-00982-7
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DOI: https://doi.org/10.1007/s10915-019-00982-7