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June 2018, 23(4): 1459-1502. doi: 10.3934/dcdsb.2018159

## Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion

 1 Department of Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan 2 MathAM-OIL, Advanced Industrial Science and Technology Tohoku, c/o AIMR, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan 3 Laboratoire de Mathématiques, University Paris-Sud Paris-Saclay and CNRS, 91405 Orsay Cedex, France

* Corresponding author: Danielle Hilhorst

Received  January 2017 Revised  January 2018 Published  April 2018

We study a time explicit finite volume method for a first order conservation law with a multiplicative source term involving a $Q$-Wiener process. After having presented the definition of a measure-valued weak entropy solution of the stochastic conservation law, we apply a finite volume method together with Godunov scheme for the space discretization, and we denote by $\{u_{\mathcal{T}, k}\}$ its discrete solution. We present some a priori estimates including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities for the discrete solution. We show that the discrete solution $\{u_{\mathcal{T}, k}\}$ converges along a subsequence to a measure-valued entropy solution of the conservation law in the sense of Young measures as the maximum diameter of the volume elements and the time step tend to zero. Some numerical simulations are presented in the case of the stochastic Burgers equation. The empirical average turns out to be a regularization of the deterministic solution; moreover, the variance in the case of the $Q$-Brownian motion converges to a constant while that in the Brownian motion case keeps increasing as time tends to infinity.

Citation: Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159
##### References:
 [1] E. Audusse, S. Boyaval, Y. Gao and D. Hilhorst, Numerical simulations of the inviscid Burgers equation with periodic boundary conditions and stochastic forcing, ESAIM: Proceedings and surveys, 48 (2015), 308-320. [2] E. J. Balder, Lectures on Young measure theory and its applications in economics, Workshop on Measure Theory and Real Analysis, Grado, 1997, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 1-69. [3] C. Bauzet, On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments, Journal of Evolution Equations, 14 (2014), 333-356. doi: 10.1007/s00028-013-0215-1. [4] C. Bauzet, G. Vallet and P. Wittbold, The Cauchy problem for a conservation law with a multiplicative stochastic perturbation, Journal of Hyperbolic Differential Equations, 9 (2012), 661-709. doi: 10.1142/S0219891612500221. [5] C. Bauzet, G. Vallet and P. Wittbold, The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation, Journal of Functional Analysis, 266 (2014), 2503-2545. doi: 10.1016/j.jfa.2013.06.022. [6] C. Bauzet, J. Charrier and T. Gallouët, Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation, Mathematics of Computation, 85 (2016), 2777-2813. doi: 10.1090/mcom/3084. [7] C. Bauzet, J. Charrier and T. Gallouët, Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with a multiplicative noise, Stochastic Partial Differential Equations: Analysis and Computations, 4 (2016), 150-223. doi: 10.1007/s40072-015-0052-z. [8] C. Bauzet, J. Charrier and T. Gallouët, Numerical approximation of stochastic conservation laws on bounded domains, Mathematical Modelling and Numerical Analysis, 51 (2017), 225-278. doi: 10.1051/m2an/2016020. [9] G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal., 204 (2012), 707-743. doi: 10.1007/s00205-011-0489-9. [10] G. Da Prato and J. Zabcyzk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Second edition, 2014. doi: 10.1017/CBO9781107295513. [11] A. Debussche and J. Vovelle, Long-time behavior in scalar conservation laws, Differential Integral Equations, 22 (2009), 225-238. [12] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, Journal of Functional Analysis, 259 (2012), 1014-1042. doi: 10.1016/j.jfa.2010.02.016. [13] R. Eymard, T. Gallouët and R. Herbin, Existence and Uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Annals of Mathematics, Series B, 16 (1995), 1-14. [14] J. Feng and D. Nualart, Stochastic scalar conservation laws, Journal of Functional Analysis, 255 (2008), 313-373. doi: 10.1016/j.jfa.2008.02.004. [15] T. Funaki, Y. Gao and D. Hilhorst, Uniqueness results for a stochastic conservation law with a Q-Brownian motion, in preparation. [16] Y. Gao, Finite Volume Methods for Deterministic and Stochastic Partial Differential Equations, Ph. D thesis, Université Paris-Sud, 2015. [17] L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011. doi: 10.1007/978-3-642-16194-0. [18] C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 1997. [19] M. Hofmanová, Bhatnagar-Gross-Krook approximation to stochastic scalar conservation laws, Ann. Inst. H. Poincaré Probab. Statist, 51 (2015), 1500-1528. doi: 10.1214/14-AIHP610. [20] H. Holden and N. H. Risebro, A stochastic approach to conservation laws, In Third International Conference on Hyperbolic Problems, 1/2 (1990), 575-587. [21] I. Kröker and C. Rohde, Finite volume schemes for hyperbolic balance laws with multiplicative noise, Appl. Number. Math., 62 (2012), 441-456. doi: 10.1016/j.apnum.2011.01.011. [22] H. H. Kuo, Introduction to Stochastic Integration, Springer, Springer Science + Business Media, Inc, 2006. [23] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Volume 68, Springer-Verlag, New York, Berlin, Heidelberg, 1988. doi: 10.1007/978-1-4684-0313-8.

show all references

##### References:
 [1] E. Audusse, S. Boyaval, Y. Gao and D. Hilhorst, Numerical simulations of the inviscid Burgers equation with periodic boundary conditions and stochastic forcing, ESAIM: Proceedings and surveys, 48 (2015), 308-320. [2] E. J. Balder, Lectures on Young measure theory and its applications in economics, Workshop on Measure Theory and Real Analysis, Grado, 1997, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 1-69. [3] C. Bauzet, On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments, Journal of Evolution Equations, 14 (2014), 333-356. doi: 10.1007/s00028-013-0215-1. [4] C. Bauzet, G. Vallet and P. Wittbold, The Cauchy problem for a conservation law with a multiplicative stochastic perturbation, Journal of Hyperbolic Differential Equations, 9 (2012), 661-709. doi: 10.1142/S0219891612500221. [5] C. Bauzet, G. Vallet and P. Wittbold, The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation, Journal of Functional Analysis, 266 (2014), 2503-2545. doi: 10.1016/j.jfa.2013.06.022. [6] C. Bauzet, J. Charrier and T. Gallouët, Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation, Mathematics of Computation, 85 (2016), 2777-2813. doi: 10.1090/mcom/3084. [7] C. Bauzet, J. Charrier and T. Gallouët, Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with a multiplicative noise, Stochastic Partial Differential Equations: Analysis and Computations, 4 (2016), 150-223. doi: 10.1007/s40072-015-0052-z. [8] C. Bauzet, J. Charrier and T. Gallouët, Numerical approximation of stochastic conservation laws on bounded domains, Mathematical Modelling and Numerical Analysis, 51 (2017), 225-278. doi: 10.1051/m2an/2016020. [9] G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal., 204 (2012), 707-743. doi: 10.1007/s00205-011-0489-9. [10] G. Da Prato and J. Zabcyzk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Second edition, 2014. doi: 10.1017/CBO9781107295513. [11] A. Debussche and J. Vovelle, Long-time behavior in scalar conservation laws, Differential Integral Equations, 22 (2009), 225-238. [12] A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, Journal of Functional Analysis, 259 (2012), 1014-1042. doi: 10.1016/j.jfa.2010.02.016. [13] R. Eymard, T. Gallouët and R. Herbin, Existence and Uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Annals of Mathematics, Series B, 16 (1995), 1-14. [14] J. Feng and D. Nualart, Stochastic scalar conservation laws, Journal of Functional Analysis, 255 (2008), 313-373. doi: 10.1016/j.jfa.2008.02.004. [15] T. Funaki, Y. Gao and D. Hilhorst, Uniqueness results for a stochastic conservation law with a Q-Brownian motion, in preparation. [16] Y. Gao, Finite Volume Methods for Deterministic and Stochastic Partial Differential Equations, Ph. D thesis, Université Paris-Sud, 2015. [17] L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011. doi: 10.1007/978-3-642-16194-0. [18] C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 1997. [19] M. Hofmanová, Bhatnagar-Gross-Krook approximation to stochastic scalar conservation laws, Ann. Inst. H. Poincaré Probab. Statist, 51 (2015), 1500-1528. doi: 10.1214/14-AIHP610. [20] H. Holden and N. H. Risebro, A stochastic approach to conservation laws, In Third International Conference on Hyperbolic Problems, 1/2 (1990), 575-587. [21] I. Kröker and C. Rohde, Finite volume schemes for hyperbolic balance laws with multiplicative noise, Appl. Number. Math., 62 (2012), 441-456. doi: 10.1016/j.apnum.2011.01.011. [22] H. H. Kuo, Introduction to Stochastic Integration, Springer, Springer Science + Business Media, Inc, 2006. [23] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Volume 68, Springer-Verlag, New York, Berlin, Heidelberg, 1988. doi: 10.1007/978-1-4684-0313-8.
Solutions in the deterministic case
The positions of the shock
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the Brownian motion case with $\alpha_B = 1/(2\pi)$ at $t = 0.05$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the Brownian motion case with $\alpha_B = 1/(2\pi)$ at $t = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the Brownian motion case with $\alpha_B = 1/(2\pi)$ at $t = 20$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 1$ and $\zeta = 1$ at $t = 0.05$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 1$ and $\zeta = 1$ at $t = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 1$ and $\zeta = 1$ at $t = 20$
Variance in the Brownian motion case (left) and in the $Q$-Brownian motion case (right) for fixed time, with $\alpha_B = 1/(2\pi)$, $\alpha_Q = 1$ and $\zeta = 1$
$L^1$ norm of the variance as a function of time in the case of Brownian motion (left) and $Q$-Brownian motion (right) with $\alpha_B = 1/(2\pi)$, $\alpha_Q = 1$ and $\zeta = 1$
$L^1$ norm of the variance as a function of time in the case of the Brownian motion with $\alpha_B = 1/(2\pi)$ (left) and $\alpha_B = 1/\pi$ (right)
Covariance in the case of Brownian motion (left) and $Q$-Brownian motion (right) as a function of time with $\alpha_Q = 1$ and $\zeta = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 2\pi$ and $\zeta = 1$ at $t = 0.05$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 2\pi$ and $\zeta = 1$ at $t = 1$
Comparing the solution in the deterministic case with the empirical average (left) and one realization (right) in the $Q$-Brownian motion case with $\alpha_Q = 2\pi$ and $\zeta = 1$ at $t = 20$
The $L^1$ norm of the variance in the cases that $\alpha_Q = 1$ (left) and $\alpha_Q = 2\pi$ (right) as a function of 10:36:59
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