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Generalized Probability Density Function of the Solution to the Random Burgers-Riemann Problem

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Numerical Fluid Dynamics

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

Abstract

In this work, we study, from a probabilistic standpoint, the Riemann problem with uncertainties for the one-dimensional inviscid Burgers equation, which comes from the Navier-Stokes equations. We assume that the initial condition is formed by two random variables. The solution to the partial differential equation problem, which is explicitly constructed by grouping together rarefaction and shock waves, becomes a random field. At each space-time location, such solution is a mixed random variable, whose generalized probability density is obtained by using simple integration and a Dirac delta function. A key point of the contribution is that no assumption on statistical independence between the random initial conditions is imposed to obtain the probability density of the solution. In order to illustrate our theoretical findings, several computational examples are included. We validate our numerical results comparing the first moments (mean and variance), calculated from the probability density function of the solution, with the results obtained via Monte Carlo simulations. Numerical comparisons show full agreement.

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References

  1. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, Course of Theoretical Physics, vol. 6. Pergamon Press, London (1959)

    Google Scholar 

  2. Quarteroni, A.: Numerical Models for Differential Problems, 2nd ed. Springer, New Delhi, India (2014)

    Google Scholar 

  3. Xiu, D.: Numerical Methods for Stochastic Computations. Princeton University Press, A Spectral Method Approach, Princeton, New Jersey (2010)

    Google Scholar 

  4. Le Maître, O., Knio, O.M.: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer Science & Business Media, Netherlands (2010)

    Google Scholar 

  5. Smith R.C.: Uncertainty Quantification: Theory, Implementation, and Applications, SIAM (2013)

    Google Scholar 

  6. Xiu, D., Karniadakis, G.E.: Supersensitivity due to uncertain boundary conditions. Int. J. Numer. Meth. Eng. 61(12), 2114–2138 (2004)

    Article  MathSciNet  Google Scholar 

  7. Pettersson, P., Iaccarino, G., Nordström, J.: Numerical analysis of the Burgers’ equation in the presence of uncertainty. J. Comput. Phys. 228(22), 8394–8412 (2009)

    Article  MathSciNet  Google Scholar 

  8. Calatayud, J., Cortés, J.-C., Jornet, M.: A modified perturbation method for mathematical models with randomness: An analysis through the steady-state solution to Burgers’ PDE. Mathematical Methods in the Applied Sciences (2020). https://doi.org/10.1002/mma.6420

  9. Kim, H.: An efficient computational method for statistical moments of Burger’s equation with random initial conditions. Math. Prob. Eng. 1–21 (2006)

    Google Scholar 

  10. Cunha, M.C.C., Dorini, F.A.: Statistical moments of the solution of the random Burgers-Riemann problem. Math. Comput. Simul. 79(5), 1440–1451 (2009)

    Article  MathSciNet  Google Scholar 

  11. Cunha, M.C.C., Dorini, F.A.: A finite volume method for the mean of the solution of the random transport equation. Appl. Math. Comput. 187(2), 912–921 (2007)

    Article  MathSciNet  Google Scholar 

  12. Cunha, M.C.C., Dorini, F.A.: A numerical scheme for the variance of the solution of the random transport equation. Appl. Math. Comput. 190(1), 362–369 (2007)

    Article  MathSciNet  Google Scholar 

  13. Cunha, M.C.C., Dorini, F.A.: A note on the Riemann problem for the random transport equation. Comput. Appl. Math. 26(3), 323–335 (2007)

    Article  MathSciNet  Google Scholar 

  14. Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216(5), 1524–1530 (2010)

    Article  MathSciNet  Google Scholar 

  15. Dorini, F.A., Cecconello, M.S., Dorini, L.B.: On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Commun. Nonlinear Sci. Numer. Simul. 33, 160–173 (2016)

    Article  MathSciNet  Google Scholar 

  16. Hussein, A., Selim, M.M.: Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Appl. Math. Comput. 218(13), 7193–7203 (2012)

    Article  MathSciNet  Google Scholar 

  17. Calatayud, J., Cortés, J.-C., Jornet, M.: Computing the density function of complex models with randomness by using polynomial expansions and the RVT technique. Appl. SIR Epidemic Model., Chaos, Solitons Fractals 133, 109639 (2020)

    Google Scholar 

  18. Calatayud, J., Cortés, J.-C., Jornet, M.: Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Math. Methods Appl. Sci. 42(17), 5649–5667 (2019)

    Article  MathSciNet  Google Scholar 

  19. Jornet, M., Calatayud, J., Le Maître, O.P., Cortés, J.-C.: Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function. J. Comput. Appl. Math. 374, 112770 (2020)

    Google Scholar 

  20. El-Tawil, M.A.: The approximate solutions of some stochastic differential equations using transformations. Appl. Math. Comput. 164, 167–178 (2005)

    Article  MathSciNet  Google Scholar 

  21. Calatayud J., Cortés J.-C., Díaz J.A., Jornet M.: Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem. Stoch: Int. J. Probab. Stoch. Process. 92(4), 627–641 (2020)

    Google Scholar 

  22. Calatayud, J., Cortés, J.-C., Díaz, J.A., Jornet, M.: Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Appl. Numer. Math. 151, 413–424 (2020)

    Article  MathSciNet  Google Scholar 

  23. Calatayud, J., Cortés, J.-C., Dorini, F.A., Jornet, M.: Extending the study on the linear advection equation subject to stochastic velocity field and initial condition. Math. Comput. Simul. 172, 159–174 (2020)

    Article  MathSciNet  Google Scholar 

  24. Cortés, J.-C., Jornet, M.: Improving kernel methods for density estimation in random differential equations problems. Math. Comput. Appl. 25(2), 33–41 (2020)

    MathSciNet  Google Scholar 

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Acknowledgements

This work has been partially supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017–89664–P. The author Marc Jornet has been supported by a postdoctoral contract from Universitat Jaume I, Spain (Acció 3.2 del Pla de Promoció de la Investigació de la Universitat Jaume I per a l’any 2020).

The authors would like to thank the reviewers and the editors for the revision of the chapter and the preparation of the book.

Author information

Authors and Affiliations

  1. Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n 46022, Valencia, Spain

    Juan Carlos Cortés

  2. Departament de Matemàtiques, Universitat Jaume I, 12071, Castellón, Spain

    Marc Jornet

Authors
  1. Juan Carlos Cortés
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  2. Marc Jornet
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Corresponding author

Correspondence to Juan Carlos Cortés .

Editor information

Editors and Affiliations

  1. School of Basic Sciences and Humanities, German Jordanian University, Amman, Jordan

    Dia Zeidan

  2. MNZ, Leipzig University of Applied Sciences, Leipzig, Germany

    Jochen Merker

  3. ISAE-ENSMA, Institut Pprime, Futuroscope Chasseneuil Cedex, France

    Eric Goncalves Da Silva

  4. Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

    Lucy T. Zhang

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Cortés, J.C., Jornet, M. (2022). Generalized Probability Density Function of the Solution to the Random Burgers-Riemann Problem. In: Zeidan, D., Merker, J., Da Silva, E.G., Zhang, L.T. (eds) Numerical Fluid Dynamics. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-16-9665-7_2

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