Error Estimates for well-balanced and time-split schemes on a locally damped wave equation

Amadori, Debora; Gosse, Laurent

doi:10.1090/mcom/3043

Error Estimates for well-balanced and time-split schemes on a locally damped wave equation
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by Debora Amadori and Laurent Gosse PDF

Math. Comp. 85 (2016), 601-633 Request permission

Abstract:

A posteriori $L^1$ error estimates are derived for both well-balanced (WB) and fractional-step (FS) numerical approximations of the unique weak solution of the Cauchy problem for the 1D semilinear damped wave equation. For setting up the WB algorithm, we proceed by rewriting it under the form of an elementary $3 \times 3$ system which linear convective structure allows to reduce the Godunov scheme with optimal Courant number (corresponding to $\Delta t=\Delta x$) to a wavefront-tracking algorithm free from any step of projection onto piecewise constant functions. A fundamental difference in the total variation estimates is proved, which partly explains the discrepancy of the FS method when the dissipative (sink) term displays an explicit dependence in the space variable. Numerical tests are performed by means of stationary exact solutions of the linear damped wave equation.

References

Saul Abarbanel, Adi Ditkowski, and Bertil Gustafsson, On error bounds of finite difference approximations to partial differential equations—temporal behavior and rate of convergence, J. Sci. Comput. 15 (2000), no. 1, 79–116. MR 1829556, DOI 10.1023/A:1007688522777
Fatiha Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, Control of partial differential equations, Lecture Notes in Math., vol. 2048, Springer, Heidelberg, 2012, pp. 1–100. MR 3220858, DOI 10.1007/978-3-642-27893-8_{1}
D. Amadori, A. Corli, Glimm estimates for a model of multiphase flow, Preprint
Debora Amadori and Laurent Gosse, Transient $L^1$ error estimates for well-balanced schemes on non-resonant scalar balance laws, J. Differential Equations 255 (2013), no. 3, 469–502. MR 3053474, DOI 10.1016/j.jde.2013.04.016
Debora Amadori and Graziano Guerra, Global BV solutions and relaxation limit for a system of conservation laws, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 1, 1–26. MR 1820292, DOI 10.1017/S0308210500000767
Debora Amadori and Graziano Guerra, Uniqueness and continuous dependence for systems of balance laws with dissipation, Nonlinear Anal. 49 (2002), no. 7, Ser. A: Theory Methods, 987–1014. MR 1895539, DOI 10.1016/S0362-546X(01)00721-0
Denise Aregba-Driollet, Maya Briani, and Roberto Natalini, Asymptotic high-order schemes for $2\times 2$ dissipative hyperbolic systems, SIAM J. Numer. Anal. 46 (2008), no. 2, 869–894. MR 2383214, DOI 10.1137/060678373
Christos Arvanitis, Theodoros Katsaounis, and Charalambos Makridakis, Adaptive finite element relaxation schemes for hyperbolic conservation laws, M2AN Math. Model. Numer. Anal. 35 (2001), no. 1, 17–33. MR 1811979, DOI 10.1051/m2an:2001105
Christos Arvanitis, Charalambos Makridakis, and Athanasios E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws, SIAM J. Numer. Anal. 42 (2004), no. 4, 1357–1393. MR 2114282, DOI 10.1137/S0036142902420436
M. Asadzadeh, D. Rostamy, and F. Zabihi, A posteriori error estimates for a coupled wave system with a local damping, J. Math. Sci. (N.Y.) 175 (2011), no. 3, 228–248. Problems in mathematical analysis. No. 56. MR 2839040, DOI 10.1007/s10958-011-0346-2
Stefano Bianchini, Bernard Hanouzet, and Roberto Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), no. 11, 1559–1622. MR 2349349, DOI 10.1002/cpa.20195
Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
Alberto Bressan, Tai-Ping Liu, and Tong Yang, $L^1$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1–22. MR 1723032, DOI 10.1007/s002050050165
Marcelo M. Cavalcanti, Flávio R. Dias Silva, and Valéria N. Domingos Cavalcanti, Uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure, SIAM J. Control Optim. 52 (2014), no. 1, 545–580. MR 3164554, DOI 10.1137/120862545
Bernardo Cockburn and Huiing Gau, A posteriori error estimates for general numerical methods for scalar conservation laws, Mat. Apl. Comput. 14 (1995), no. 1, 37–47 (English, with English and Portuguese summaries). MR 1339916
Frédéric Coquel, Shi Jin, Jian-Guo Liu, and Li Wang, Well-posedness and singular limit of a semilinear hyperbolic relaxation system with a two-scale discontinuous relaxation rate, Arch. Ration. Mech. Anal. 214 (2014), no. 3, 1051–1084. MR 3269642, DOI 10.1007/s00205-014-0773-6
Rinaldo M. Colombo and Graziano Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst. 23 (2009), no. 3, 733–753. MR 2461824, DOI 10.3934/dcds.2009.23.733
Michael Crandall and Andrew Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), no. 3, 285–314. MR 571291, DOI 10.1007/BF01396704
J. Glimm and D. H. Sharp, An $S$ matrix theory for classical nonlinear physics, Found. Phys. 16 (1986), no. 2, 125–141. MR 836850, DOI 10.1007/BF01889377
Laurent Gosse, Computing qualitatively correct approximations of balance laws, SIMAI Springer Series, vol. 2, Springer, Milan, 2013. Exponential-fit, well-balanced and asymptotic-preserving. MR 3053000, DOI 10.1007/978-88-470-2892-0
Laurent Gosse and Charalambos Makridakis, Two a posteriori error estimates for one-dimensional scalar conservation laws, SIAM J. Numer. Anal. 38 (2000), no. 3, 964–988. MR 1781211, DOI 10.1137/S0036142999350383
Laurent Gosse and Giuseppe Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 337–342 (English, with English and French summaries). MR 1891014, DOI 10.1016/S1631-073X(02)02257-4
Mark Kac, A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math. 4 (1974), 497–509. Reprinting of an article published in 1956. MR 510166, DOI 10.1216/RMJ-1974-4-3-497
D. Kaya, An application of the decomposition method for second order wave equations, Int. J. Comput. Math. 75 (2000), no. 2, 235–245. MR 1787281, DOI 10.1080/00207160008804979
H. Kim, M. Laforest, and D. Yoon, An adaptive version of Glimm’s scheme, Acta Math. Sci. Ser. B (Engl. Ed.) 30 (2010), no. 2, 428–446. MR 2656549, DOI 10.1016/S0252-9602(10)60057-4
M. Laforest, A posteriori error estimate for front-tracking: systems of conservation laws, SIAM J. Math. Anal. 35 (2004), no. 5, 1347–1370. MR 2050203, DOI 10.1137/S0036141002416870
Wei-Jiu Liu and Enrique Zuazua, Decay rates for dissipative wave equations, Ricerche Mat. 48 (1999), no. suppl., 61–75. Papers in memory of Ennio De Giorgi (Italian). MR 1765677
Brian J. McCartin, Exponential fitting of the damped wave equation, Appl. Math. Sci. (Ruse) 4 (2010), no. 37-40, 1899–1930. MR 2670064
Pierangelo Marcati and Kenji Nishihara, The $L^p$-$L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations 191 (2003), no. 2, 445–469. MR 1978385, DOI 10.1016/S0022-0396(03)00026-3
Arnaud Münch and Ademir Fernando Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation, ESAIM Control Optim. Calc. Var. 13 (2007), no. 2, 265–293. MR 2306636, DOI 10.1051/cocv:2007009
Mario Ohlberger, A review of a posteriori error control and adaptivity for approximations of non-linear conservation laws, Internat. J. Numer. Methods Fluids 59 (2009), no. 3, 333–354. MR 2484271, DOI 10.1002/fld.1686
Gabriella Puppo and Matteo Semplice, Numerical entropy and adaptivity for finite volume schemes, Commun. Comput. Phys. 10 (2011), no. 5, 1132–1160. MR 2830842, DOI 10.4208/cicp.250909.210111a
Jeffrey Rauch and Michael Reed, Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: creation and propagation, Comm. Math. Phys. 81 (1981), no. 2, 203–227. MR 632757
Louis Roder Tcheugoué Tébou and Enrique Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numer. Math. 95 (2003), no. 3, 563–598. MR 2012934, DOI 10.1007/s00211-002-0442-9
Judith Vancostenoble, Optimalité d’estimations d’énergie pour une équation des ondes amortie, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 9, 777–782 (French, with English and French summaries). MR 1694086, DOI 10.1016/S0764-4442(99)80271-7
T. Vejchodsky, On A-Posteriori error estimation strategies for elliptic problems, in J. Privratska, J. Prihonska, D. Andrejsova (Eds.), Proceedings of International Conference ICPM’05, Liberec, Czech Republic (2005), 373–386.
R. Verfurth, A review of a posteriori error estimation and adaptive mesh refinement techniques, Teubner-Wiley (1996)