Bauzet, C.; Charrier, J.; Gallouët, T. Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation. (English) Zbl 1351.65003 Math. Comput. 85, No. 302, 2777-2813 (2016). Summary: Here, we study explicit flux-splitting finite volume discretizations of multi-dimensional nonlinear scalar conservation laws perturbed by a multiplicative noise with a given initial data in \( L^{2}(\mathbb{R}^d)\). Under a stability condition on the time step, we prove the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation. Cited in 11 Documents MSC: 65C30 Numerical solutions to stochastic differential and integral equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) Keywords:stochastic PDE; first-order hyperbolic equation; Itô integral; multiplicative noise; finite volume method; flux-splitting scheme; Engquist-Osher scheme; Lax-Friedrichs scheme; upwind scheme; Young measures; Kruzhkov smooth entropy; stability; convergence; stochastic entropy solution PDFBibTeX XMLCite \textit{C. Bauzet} et al., Math. Comput. 85, No. 302, 2777--2813 (2016; Zbl 1351.65003) Full Text: DOI HAL References: [1] Balder, Erik J., Lectures on Young measure theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste, 31, suppl. 1, 1-69 (2000) · Zbl 1032.91007 [2] Bauzet, Caroline, On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments, J. Evol. Equ., 14, 2, 333-356 (2014) · Zbl 1295.35313 · doi:10.1007/s00028-013-0215-1 [3] Bauzet, Caroline; Vallet, Guy; Wittbold, Petra, The Cauchy problem for conservation laws with a multiplicative stochastic perturbation, J. Hyperbolic Differ. Equ., 9, 4, 661-709 (2012) · Zbl 1263.35222 · doi:10.1142/S0219891612500221 [4] Bauzet, Caroline; Vallet, Guy; Wittbold, Petra, The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation, J. Funct. Anal., 266, 4, 2503-2545 (2014) · Zbl 1292.35168 · doi:10.1016/j.jfa.2013.06.022 [5] Chen, Gui-Qiang; Ding, Qian; Karlsen, Kenneth H., On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal., 204, 3, 707-743 (2012) · Zbl 1261.60062 · doi:10.1007/s00205-011-0489-9 [6] Champier, S.; Gallou{\"e}t, T.; Herbin, R., Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh, Numer. Math., 66, 2, 139-157 (1993) · Zbl 0801.65089 · doi:10.1007/BF01385691 [7] Da Prato, Giuseppe; Zabczyk, Jerzy, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications 44, xviii+454 pp. (1992), Cambridge University Press, Cambridge · Zbl 1140.60034 · doi:10.1017/CBO9780511666223 [8] Debussche, A.; Vovelle, J., Scalar conservation laws with stochastic forcing, J. Funct. Anal., 259, 4, 1014-1042 (2010) · Zbl 1200.60050 · doi:10.1016/j.jfa.2010.02.016 [9] Eymard, R.; Gallou{\"e}t, T.; Herbin, R., Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Ann. Math. Ser. B, 16, 1, 1-14 (1995) · Zbl 0830.35077 [10] Eymard, Robert; Gallou{\"e}t, Thierry; Herbin, Rapha{\`e}le, Finite Volume Methods, Handb. Numer. Anal., VII, 713-1020 (2000), North-Holland, Amsterdam · Zbl 0981.65095 [11] Feng, Jin; Nualart, David, Stochastic scalar conservation laws, J. Funct. Anal., 255, 2, 313-373 (2008) · Zbl 1154.60052 · doi:10.1016/j.jfa.2008.02.004 [12] Hofmanov{\'a}, Martina, A Bhatnagar-Gross-Krook approximation to stochastic scalar conservation laws, Ann. Inst. Henri Poincar\'e Probab. Stat., 51, 4, 1500-1528 (2015) · Zbl 1329.60214 · doi:10.1214/14-AIHP610 [13] Holden, H.; Risebro, N. H., A stochastic approach to conservation laws. Third International Conference on Hyperbolic Problems, Vol.I, II , Uppsala, 1990, 575-587 (1991), Studentlitteratur, Lund · Zbl 0789.35103 [14] Kim, Jong Uhn, On the stochastic porous medium equation, J. Differential Equations, 220, 1, 163-194 (2006) · Zbl 1099.35187 · doi:10.1016/j.jde.2005.02.006 [15] Kr{\"o}ker, I.; Rohde, C., Finite volume schemes for hyperbolic balance laws with multiplicative noise, Appl. Numer. Math., 62, 4, 441-456 (2012) · Zbl 1241.65013 · doi:10.1016/j.apnum.2011.01.011 [16] Kr{\'”o}ker, Ilja, Finite volume methods for conservation laws with noise. Finite volumes for complex applications V, 527-534 (2008), ISTE, London · Zbl 1422.65209 [17] Panov, E. Yu., On measure-valued solutions of the Cauchy problem for a first-order quasilinear equation, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 60 60, 2, 335-377 (1996) · Zbl 0882.35075 · doi:10.1070/IM1996v060n02ABEH000073 [18] Vallet, G., Stochastic perturbation of nonlinear degenerate parabolic problems, Differential Integral Equations, 21, 11-12, 1055-1082 (2008) · Zbl 1224.35218 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.