• Previous Article
    An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems
  • DCDS-B Home
  • This Issue
  • Next Article
    A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model
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. AudusseS. BoyavalY. 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

C. BauzetG. 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.  Google Scholar

[5]

C. BauzetG. 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.  Google Scholar

[6]

C. BauzetJ. 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.  Google Scholar

[7]

C. BauzetJ. 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.  Google Scholar

[8]

C. BauzetJ. 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.  Google Scholar

[9]

G.-Q. ChenQ. 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.  Google Scholar

[10] G. Da Prato and J. Zabcyzk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Second edition, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[11]

A. Debussche and J. Vovelle, Long-time behavior in scalar conservation laws, Differential Integral Equations, 22 (2009), 225-238.   Google Scholar

[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.  Google Scholar

[13]

R. EymardT. 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.   Google Scholar

[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.  Google Scholar

[15]

T. Funaki, Y. Gao and D. Hilhorst, Uniqueness results for a stochastic conservation law with a Q-Brownian motion, in preparation. Google Scholar

[16]

Y. Gao, Finite Volume Methods for Deterministic and Stochastic Partial Differential Equations, Ph. D thesis, Université Paris-Sud, 2015. Google Scholar

[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.  Google Scholar
[18]

C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 1997. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

H. H. Kuo, Introduction to Stochastic Integration, Springer, Springer Science + Business Media, Inc, 2006.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

E. AudusseS. BoyavalY. 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

C. BauzetG. 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.  Google Scholar

[5]

C. BauzetG. 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.  Google Scholar

[6]

C. BauzetJ. 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.  Google Scholar

[7]

C. BauzetJ. 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.  Google Scholar

[8]

C. BauzetJ. 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.  Google Scholar

[9]

G.-Q. ChenQ. 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.  Google Scholar

[10] G. Da Prato and J. Zabcyzk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Second edition, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[11]

A. Debussche and J. Vovelle, Long-time behavior in scalar conservation laws, Differential Integral Equations, 22 (2009), 225-238.   Google Scholar

[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.  Google Scholar

[13]

R. EymardT. 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.   Google Scholar

[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.  Google Scholar

[15]

T. Funaki, Y. Gao and D. Hilhorst, Uniqueness results for a stochastic conservation law with a Q-Brownian motion, in preparation. Google Scholar

[16]

Y. Gao, Finite Volume Methods for Deterministic and Stochastic Partial Differential Equations, Ph. D thesis, Université Paris-Sud, 2015. Google Scholar

[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.  Google Scholar
[18]

C. M. Grinstead and J. L. Snell, Introduction to Probability, American Mathematical Society, 1997. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

H. H. Kuo, Introduction to Stochastic Integration, Springer, Springer Science + Business Media, Inc, 2006.  Google Scholar

[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.  Google Scholar

Figure 1.  Solutions in the deterministic case
Figure 2.  The positions of the shock
Figure 3.  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 $t = 0.05$
Figure 4.  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 $t = 1$
Figure 5.  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 $t = 20$
Figure 6.  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 $t = 0.05$
Figure 7.  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 $t = 1$
Figure 8.  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 $t = 20$
Figure 9.  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 $\zeta = 1$
Figure 10.  $L^1$ 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 $\zeta = 1$
Figure 11.  $L^1$ 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)
Figure 12.  Covariance in the case of Brownian motion (left) and $Q$-Brownian motion (right) as a function of time with $\alpha_Q = 1$ and $\zeta = 1$
Figure 13.  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 $t = 0.05$
Figure 14.  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 $t = 1$
Figure 15.  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 $t = 20$
Figure 16.  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
[1]

Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283

[2]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

[3]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004

[4]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289

[5]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561

[6]

Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427

[7]

Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269

[8]

Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188

[9]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[10]

Emmanuel Frénod. Homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i

[11]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[12]

Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215

[13]

Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15

[14]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233

[15]

B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463

[16]

Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669

[17]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[18]

Emmanuel Frénod. An attempt at classifying homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : i-vi. doi: 10.3934/dcdss.2015.8.1i

[19]

Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220

[20]

Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (69)
  • HTML views (165)
  • Cited by (0)

Other articles
by authors

[Back to Top]