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Convergence of the finite volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation. (English) Zbl 1446.65089

Summary: Under a standard CFL condition, we prove the convergence of the explicit-in-time Finite Volume method with monotone fluxes for the approximation of scalar first-order conservation laws with multiplicative, compactly supported noise. In [S. Dotti and J. Vovelle, Arch. Ration. Mech. Anal. 230, No. 2, 539–591 (2018; Zbl 1397.65016)], a framework for the analysis of the convergence of approximations to stochastic scalar first-order conservation laws has been developed, on the basis of a kinetic formulation. Here, we give a kinetic formulation of the numerical method, analyse its properties, and explain how to cast the problem of convergence of the numerical scheme into the framework of Dotti and Vovelle [loc. cit.]. This uses standard estimates (like the so-called “weak BV estimate”, for which we give a proof using specifically the kinetic formulation) and an adequate interpolation procedure.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65C30 Numerical solutions to stochastic differential and integral equations
60G57 Random measures

Citations:

Zbl 1397.65016
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References:

[1] Bauzet, C., Time-splitting approximation of the Cauchy problem for a stochastic conservation law, Math. Comput. Simul., 118, 73-86 (2015)
[2] Bauzet, C.; Vallet, G.; Wittbold, P., The Cauchy problem for conservation laws with a multiplicative stochastic perturbation, J. Hyperb. Differ. Equ., 9, 4, 661-709 (2012) · Zbl 1263.35222
[3] Bauzet, C.; Vallet, G.; Wittbold, P., The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation, J. Funct. Anal., 266, 4, 2503-2545 (2014) · Zbl 1292.35168
[4] Bauzet, C.; Charrier, J.; Gallouët, T., Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation, Math. Comput., 85, 302, 2777-2813 (2016) · Zbl 1351.65003
[5] Bauzet, C.; Charrier, J.; Gallouët, T., Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise, Stoch. Partial Differ. Equ. Anal. Comput., 4, 1, 150-223 (2016) · Zbl 1358.65007
[6] Chen, G-Q; Ding, Q.; Karlsen, KH, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal., 204, 3, 707-743 (2012) · Zbl 1261.60062
[7] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications (1992), Cambridge: Cambridge University Press, Cambridge
[8] Debussche, A.; Vovelle, J., Scalar conservation laws with stochastic forcing, J. Funct. Anal., 259, 4, 1014-1042 (2010) · Zbl 1200.60050
[9] Dotti, S.; Vovelle, J., Convergence of approximations to stochastic scalar conservation laws, Arch. Ration. Mech. Anal., 230, 2, 539-591 (2018) · Zbl 1397.65016
[10] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, pages 713-1020. North-Holland, Amsterdam (2000) · Zbl 0981.65095
[11] Feng, J.; Nualart, D., Stochastic scalar conservation laws, J. Funct. Anal., 255, 2, 313-373 (2008) · Zbl 1154.60052
[12] Gess, B.; Souganidis, PE, Scalar conservation laws with multiple rough fluxes, Commun. Math. Sci., 13, 6, 1569-1597 (2015) · Zbl 1329.60210
[13] Gess, B.; Perthame, B.; Souganidis, PE, Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes, SIAM J. Numer. Anal., 54, 4, 2187-2209 (2016) · Zbl 1345.60069
[14] Gess, B.; Souganidis, PE, Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws, Comm. Pure Appl. Math., 70, 8, 1562-1597 (2017) · Zbl 1370.60105
[15] Hofmanová, M., Scalar conservation laws with rough flux and stochastic forcing. Stochastic partial differential equations, Anal. Comput., 4, 3, 635-690 (2016) · Zbl 1351.60080
[16] Holden, H., Risebro, N. H.: A stochastic approach to conservation laws. In Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990), pages 575-587. Studentlitteratur, Lund (1991) · Zbl 0789.35103
[17] Karlsen, KH; Storrøsten, EB, On stochastic conservation laws and Malliavin calculus, J. Funct. Anal., 272, 2, 421-497 (2017) · Zbl 1354.60070
[18] Kim, Y., Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to \(N\)-waves, J. Differ. Equ., 192, 1, 202-224 (2003) · Zbl 1026.35066
[19] Koley, U.; Majee, AK; Vallet, G., A finite difference scheme for conservation laws driven by Levy noise, IMA J. Numer. Anal., 38, 2, 998-1050 (2018) · Zbl 1477.65137
[20] Kröker, I.; Rohde, C., Finite volume schemes for hyperbolic balance laws with multiplicative noise, Appl. Numer. Math., 62, 4, 441-456 (2012) · Zbl 1241.65013
[21] Lions, P.-L., Perthame, B., Souganidis, P.E.: Stochastic averaging lemmas for kinetic equations. In Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2011-2012, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXVI, 17. École Polytech., Palaiseau (2013) · Zbl 1323.35234
[22] Lions, P-L; Perthame, B.; Tadmor, E., A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7, 1, 169-191 (1994) · Zbl 0820.35094
[23] Lions, P-L; Perthame, B.; Tadmor, E., Kinetic formulation of the isentropic gas dynamics and \(p\)-systems, Commun. Math. Phys., 163, 2, 415-431 (1994) · Zbl 0799.35151
[24] Lions, P-L; Perthame, B.; Souganidis, PE, Scalar conservation laws with rough (stochastic) fluxes, Stoch. Partial Differ. Equ. Anal. Comput., 1, 4, 664-686 (2013) · Zbl 1333.60140
[25] Lions, P-L; Perthame, B.; Souganidis, PE, Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case, Stoch. Partial Differ. Equ. Anal. Comput., 2, 4, 517-538 (2014) · Zbl 1323.35233
[26] Makridakis, C.; Perthame, B., Sharp CFL, discrete kinetic formulation, and entropic schemes for scalar conservation laws, SIAM J. Numer. Anal., 41, 3, 1032-1051 (2003) · Zbl 1066.65086
[27] Mohamed, K.; Seaid, M.; Zahri, M., A finite volume method for scalar conservation laws with stochastic time-space dependent flux functions, J. Comput. Appl. Math., 237, 1, 614-632 (2013) · Zbl 1252.65152
[28] Perthame, B.: Kinetic formulation of conservation laws, volume 21 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002) · Zbl 1030.35002
[29] Perthame, B., Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure., J. Math. Pures Appl. (9), 77, 10, 1055-1064 (1998) · Zbl 0919.35088
[30] Schilling, R. L., Partzsch, L.: Brownian Motion. De Gruyter Graduate. De Gruyter, Berlin, second edition, (2014). An introduction to stochastic processes, With a chapter on simulation by Björn Böttcher · Zbl 1283.60003
[31] Vallet, G.; Wittbold, P., On a stochastic first-order hyperbolic equation in a bounded domain, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12, 4, 613-651 (2009) · Zbl 1194.60042
[32] Weinan, E.; Khanin, K.; Mazel, A.; Sinai, Y., Invariant measures for Burgers equation with stochastic forcing, Ann. Math., 151, 3, 877-960 (2000) · Zbl 0972.35196
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