Abstract
In this paper, we are concerned with an operator-splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Lévy noise. More specifically, using a variant of the classical Kružkov doubling of variables approach, we show that the approximate solutions generated by the splitting scheme converge to the unique stochastic entropy solution of the underlying problems. Finally, the convergence analysis is illustrated by several numerical examples.
Award Identifier / Grant number: IFA18-MA119
Funding statement: The first author would like to acknowledge the financial support of CSIR, India. The second author is supported by the Department of Science and Technology, Govt. of India – the INSPIRE fellowship (IFA18-MA119).
Acknowledgements
The authors wish to thank Prof. Ujjwal Koley for his valuable discussions and suggestions.
References
[1] N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ. 7 (2007), no. 1, 145–175. Search in Google Scholar
[2] N. Alibaud, S. Cifani and E. R. Jakobsen, Continuous dependence estimates for nonlinear fractional convection-diffusion equations, SIAM J. Math. Anal. 44 (2012), no. 2, 603–632. Search in Google Scholar
[3] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Stud. Adv. Math. 116, Cambridge University, Cambridge, 2009. Search in Google Scholar
[4] E. J. Balder, Lectures on Young measure theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste 31 (2000), 1–69. Search in Google Scholar
[5] C. Bauzet, Time-splitting approximation of the Cauchy problem for a stochastic conservation law, Math. Comput. Simulation 118 (2015), 73–86. Search in Google Scholar
[6] C. Bauzet, V. Castel and J. Charrier, Existence and uniqueness result for an hyperbolic scalar conservation law with a stochastic force using a finite volume approximation, J. Hyperbolic Differ. Equ. 17 (2020), no. 2, 213–294. Search in Google Scholar
[7] C. Bauzet, J. Charrier and T. Gallouët, Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation, Math. Comp. 85 (2016), no. 302, 2777–2813. Search in Google Scholar
[8] C. Bauzet, J. Charrier and T. Gallouët, Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 1, 150–223. Search in Google Scholar
[9] C. Bauzet, G. Vallet and P. Wittbold, The Cauchy problem for conservation laws with a multiplicative stochastic perturbation, J. Hyperbolic Differ. Equ. 9 (2012), no. 4, 661–709. Search in Google Scholar
[10] C. Bauzet, G. Vallet and P. Wittbold, A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force, J. Hyperbolic Differ. Equ. 12 (2015), no. 3, 501–533. Search in Google Scholar
[11] N. Bhauryal, U. Koley and G. Vallet, The Cauchy problem for fractional conservation laws driven by Lévy noise, Stochastic Process. Appl. 130 (2020), no. 9, 5310–5365. Search in Google Scholar
[12] N. Bhauryal, U. Koley and G. Vallet, A fractional degenerate parabolic-hyperbolic Cauchy problem with noise, J. Differential Equations 284 (2021), 433–521. Search in Google Scholar
[13] I. H. Biswas, K. H. Karlsen and A. K. Majee, Conservation laws driven by Lévy white noise, J. Hyperbolic Differ. Equ. 12 (2015), no. 3, 581–654. Search in Google Scholar
[14] I. H. Biswas, U. Koley and A. K. Majee, Continuous dependence estimate for conservation laws with Lévy noise, J. Differential Equations 259 (2015), no. 9, 4683–4706. Search in Google Scholar
[15] I. H. Biswas and A. K. Majee, Stochastic conservation laws: Weak-in-time formulation and strong entropy condition, J. Funct. Anal. 267 (2014), no. 7, 2199–2252. Search in Google Scholar
[16] I. H. Biswas, A. K. Majee and G. Vallet, On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise, Adv. Nonlinear Anal. 8 (2019), no. 1, 809–844. Search in Google Scholar
[17] A. Chaudhary, Stochastic degenerate fractional conservation laws, NoDEA Nonlinear Differential Equations Appl. 30 (2023), no. 3, Paper No. 42. Search in Google Scholar
[18] A. Chaudhary, Stochastic fractional conservation laws, J. Math. Anal. Appl. 531 (2024), no. 1, Paper No. 127752. Search in Google Scholar
[19] G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal. 204 (2012), no. 3, 707–743. Search in Google Scholar
[20] P.-L. Chow, Stochastic Partial Differential Equations, 2nd ed., Chapman & Hall/CRC Appl. Math. Nonlinear Sci., Chapman & Hall/CRC, Boca Raton, 2014. Search in Google Scholar
[21] S. Cifani and E. 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. Search in Google Scholar
[22] S. Cifani and E. R. Jakobsen, On numerical methods and error estimates for degenerate fractional convection-diffusion equations, Numer. Math. 127 (2014), no. 3, 447–483. Search in Google Scholar
[23] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. 325, Springer, Berlin, 2000. Search in Google Scholar
[24] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University, Cambridge, 2014. Search in Google Scholar
[25] A. Debussche, M. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab. 44 (2016), no. 3, 1916–1955. Search in Google Scholar
[26] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259 (2010), no. 4, 1014–1042. Search in Google Scholar
[27] Z. Dong and T. G. Xu, One-dimensional stochastic Burgers equation driven by Lévy processes, J. Funct. Anal. 243 (2007), no. 2, 631–678. Search in Google Scholar
[28] S. Dotti and J. Vovelle, Convergence of approximations to stochastic scalar conservation laws, Arch. Ration. Mech. Anal. 230 (2018), no. 2, 539–591. Search in Google Scholar
[29] S. Dotti and J. Vovelle, Convergence of the finite volume method for scalar conservation laws with multiplicative noise: An approach by kinetic formulation, Stoch. Partial Differ. Equ. Anal. Comput. 8 (2020), no. 2, 265–310. Search in Google Scholar
[30]
J. Endal and E. R. Jakobsen,
[31] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar
[32] J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Funct. Anal. 255 (2008), no. 2, 313–373. Search in Google Scholar
[33] E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Math. Appl. (Paris) 3/4, Ellipses, Paris, 1991. Search in Google Scholar
[34] H. Holden and N. H. Risebro, Conservation laws with a random source, Appl. Math. Optim. 36 (1997), no. 2, 229–241. Search in Google Scholar
[35] K. H. Karlsen and E. B. Storrø sten, On stochastic conservation laws and Malliavin calculus, J. Funct. Anal. 272 (2017), no. 2, 421–497. Search in Google Scholar
[36] K. H. Karlsen and E. B. Storrø sten, Analysis of a splitting method for stochastic balance laws, IMA J. Numer. Anal. 38 (2018), no. 1, 1–56. Search in Google Scholar
[37] J. U. Kim, On a stochastic scalar conservation law, Indiana Univ. Math. J. 52 (2003), no. 1, 227–256. Search in Google Scholar
[38] U. Koley, A. K. Majee and G. Vallet, Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Lévy noise, Stoch. Partial Differ. Equ. Anal. Comput. 5 (2017), no. 2, 145–191. Search in Google Scholar
[39] U. Koley, A. K. Majee and G. Vallet, A finite difference scheme for conservation laws driven by Lévy noise, IMA J. Numer. Anal. 38 (2018), no. 2, 998–1050. Search in Google Scholar
[40] U. Koley, D. Ray and T. Sarkar, Multilevel Monte Carlo finite difference methods for fractional conservation laws with random data, SIAM/ASA J. Uncertain. Quantif. 9 (2021), 65–105. Search in Google Scholar
[41] U. Koley and G. Vallet, On the rate of convergence of a numerical scheme for fractional conservation laws with noise, IMA J. Numer. Anal. (2023), 10.1093/imanum/drad015. 10.1093/imanum/drad015Search in Google Scholar
[42] I. Kröker and C. Rohde, Finite volume schemes for hyperbolic balance laws with multiplicative noise, Appl. Numer. Math. 62 (2012), no. 4, 441–456. Search in Google Scholar
[43] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N. S.) 81(123) (1970), 228–255. Search in Google Scholar
[44] A. K. Majee, Convergence of a flux-splitting finite volume scheme for conservation laws driven by Lévy noise, Appl. Math. Comput. 338 (2018), 676–697. Search in Google Scholar
[45] A. I. Volpert, Generalized solutions of degenerate second-order quasilinear parabolic and elliptic equations, Adv. Differential Equations 5 (2000), no. 10–12, 1493–1518. Search in Google Scholar
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