On the spectral vanishing viscosity method for periodic fractional conservation laws
HTML articles powered by AMS MathViewer
- by Simone Cifani and Espen R. Jakobsen
- Math. Comp. 82 (2013), 1489-1514
- DOI: https://doi.org/10.1090/S0025-5718-2013-02690-7
- Published electronically: March 19, 2013
- PDF | Request permission
Abstract:
We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump Lévy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kružkov-Alibaud entropy formulation and prove well-posedness. Then we introduce the numerical method, which is a non-linear Fourier Galerkin method with an additional spectral viscosity term. This type of approximation was first introduced by Tadmor for pure conservation laws. We prove that this non-monotone method converges to the entropy solution of the problem, that it retains the spectral accuracy of the Fourier method, and that it diagonalizes the fractional term reducing dramatically the computational cost induced by this term. We also derive a robust $L^1$-error estimate, and provide numerical experiments for the fractional Burgers’ equation.References
- Nathaël Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ. 7 (2007), no. 1, 145–175. MR 2305729, DOI 10.1007/s00028-006-0253-z
- Nathaël Alibaud, Jérôme Droniou, and Julien Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 479–499. MR 2339805, DOI 10.1142/S0219891607001227
- David Applebaum, Lévy processes and stochastic calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009. MR 2512800, DOI 10.1017/CBO9780511809781
- Ben-Yu Guo, Spectral methods and their applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. MR 1641586, DOI 10.1142/9789812816641
- Piotr Biler, Tadahisa Funaki, and Wojbor A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), no. 1, 9–46. MR 1637513, DOI 10.1006/jdeq.1998.3458
- C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, Scientific Computation, Springer-Verlag, Berlin, 2006. Fundamentals in single domains. MR 2223552
- Chi Hin Chan and Magdalena Czubak, Regularity of solutions for the critical $N$-dimensional Burgers’ equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 2, 471–501 (English, with English and French summaries). MR 2595188, DOI 10.1016/j.anihpc.2009.11.008
- Chi Hin Chan, Magdalena Czubak, and Luis Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 847–861. MR 2600693, DOI 10.3934/dcds.2010.27.847
- Hongjie Dong, Dapeng Du, and Dong Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J. 58 (2009), no. 2, 807–821. MR 2514389, DOI 10.1512/iumj.2009.58.3505
- Gui Qiang Chen, Qiang Du, and Eitan Tadmor, Spectral viscosity approximations to multidimensional scalar conservation laws, Math. Comp. 61 (1993), no. 204, 629–643. MR 1185240, DOI 10.1090/S0025-5718-1993-1185240-3
- S. Cifani. On nonlinear fractional convection-diffusion equations. PhD Thesis 2011:282, NTNU, 2011.
- Simone Cifani and Espen R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 3, 413–441. MR 2795714, DOI 10.1016/j.anihpc.2011.02.006
- Simone Cifani, Espen R. Jakobsen, and Kenneth H. Karlsen, The discontinuous Galerkin method for fractal conservation laws, IMA J. Numer. Anal. 31 (2011), no. 3, 1090–1122. MR 2832791, DOI 10.1093/imanum/drq006
- P. Clavin. Instabilities and nonlinear patterns of overdriven detonations in gases. Nonlinear PDE’s in Condensed Matter and Reactive Flows. Kluwer, 49–97, 2002.
- Rama Cont and Peter Tankov, Financial modelling with jump processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2042661
- Jérôme Droniou, A numerical method for fractal conservation laws, Math. Comp. 79 (2010), no. 269, 95–124. MR 2552219, DOI 10.1090/S0025-5718-09-02293-5
- J. Droniou, T. Gallouet, and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ. 3 (2003), no. 3, 499–521. Dedicated to Philippe Bénilan. MR 2019032, DOI 10.1007/s00028-003-0503-1
- Jérôme Droniou and Cyril Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal. 182 (2006), no. 2, 299–331. MR 2259335, DOI 10.1007/s00205-006-0429-2
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- Andreas Dedner and Christian Rohde, Numerical approximation of entropy solutions for hyperbolic integro-differential equations, Numer. Math. 97 (2004), no. 3, 441–471. MR 2059465, DOI 10.1007/s00211-003-0502-9
- Maria Giovanna Garroni and Jose Luis Menaldi, Second order elliptic integro-differential problems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 430, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1911531, DOI 10.1201/9781420035797
- Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR 1912206, DOI 10.1007/978-3-642-56139-9
- Harold Jeffreys and Bertha Swirles, Methods of mathematical physics, Cambridge University Press, Cambridge, 1999. Reprint of the third (1956) edition. MR 1744997
- Kenneth H. Karlsen and Süleyman Ulusoy, Stability of entropy solutions for Lévy mixed hyperbolic-parabolic equations, Electron. J. Differential Equations (2011), No. 116, 23. MR 2836797
- Alexander Kiselev, Fedor Nazarov, and Roman Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008), no. 3, 211–240. MR 2455893, DOI 10.4310/DPDE.2008.v5.n3.a2
- S. N. Kružkov. First order quasi-linear equations in several independent variables. Math. USSR Sbornik, 10(2):217–243, 1970.
- N. N. Kuznetsov. Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR. Comput. Math. Phys. 16 (1976), 105–119.
- P. Loya. Dirichlet and Fresnel integrals via iterated integration. Mathematics Magazine 78 (2005), no. 1, 63–67.
- Yvon Maday and Eitan Tadmor, Analysis of the spectral vanishing viscosity method for periodic conservation laws, SIAM J. Numer. Anal. 26 (1989), no. 4, 854–870. MR 1005513, DOI 10.1137/0726047
- J. Málek, J. Nečas, M. Rokyta, and M. Růžička, Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computation, vol. 13, Chapman & Hall, London, 1996. MR 1409366, DOI 10.1007/978-1-4899-6824-1
- K. Sato. Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, 1999.
- S. Schochet, The rate of convergence of spectral-viscosity methods for periodic scalar conservation laws, SIAM J. Numer. Anal. 27 (1990), no. 5, 1142–1159. MR 1061123, DOI 10.1137/0727066
- Eitan Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal. 26 (1989), no. 1, 30–44. MR 977947, DOI 10.1137/0726003
- Eitan Tadmor, Total variation and error estimates for spectral viscosity approximations, Math. Comp. 60 (1993), no. 201, 245–256. MR 1153170, DOI 10.1090/S0025-5718-1993-1153170-9
- Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
- Wojbor A. Woyczyński, Lévy processes in the physical sciences, Lévy processes, Birkhäuser Boston, Boston, MA, 2001, pp. 241–266. MR 1833700
Bibliographic Information
- Simone Cifani
- Affiliation: Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
- Email: simone.cifani@math.ntnu.no
- Espen R. Jakobsen
- Affiliation: Department of Mathematics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
- Email: erj@math.ntnu.no
- Received by editor(s): November 15, 2010
- Received by editor(s) in revised form: November 29, 2011
- Published electronically: March 19, 2013
- Additional Notes: This research was supported by the Research Council of Norway (NFR) through the project “Integro-PDEs: Numerical Methods, Analysis, and Applications to Finance”.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 1489-1514
- MSC (2010): Primary 65M70, 35K59, 35R09; Secondary 65M15, 65M12, 35K57, 35R11
- DOI: https://doi.org/10.1090/S0025-5718-2013-02690-7
- MathSciNet review: 3042572