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Numerical methods for conservation laws with rough flux

  • H. HoelEmail author
  • K. H. Karlsen
  • N. H. Risebro
  • E. B. Storrøsten
Article
  • 3 Downloads

Abstract

Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for \(\alpha \)-Hölder continuous paths the convergence rate of the numerical methods can improve from \(\mathcal {O}(\text {COST}^{-\gamma })\), for some \(\gamma \in \left[ \alpha /(12-8\alpha ), \alpha /(10-6\alpha )\right] \), with \(\alpha \in (0, 1)\), to \(\mathcal {O}(\text {COST}^{-\min (1/4,\alpha /2)})\). Numerical examples support the theoretical results.

Keywords

Stochastic conservation law Rough time-dependent flux Pathwise entropy solution Finite difference method Convergence Stochastic numerics 

Mathematics Subject Classification

Primary: 35L65 65M06 Secondary: 60H15 65C30 

Notes

References

  1. 1.
    Abgrall, R., Mishra, S.: Uncertainty quantification for hyperbolic systems of conservation laws. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems. Handbook of Numerical Analysis vol. 18, pp. 507–544. Elsevier, (2017)Google Scholar
  2. 2.
    Attanasio, S., Flandoli, F.: Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplication noise. Commun. Partial Differ. Equ. 36(8), 1455–1474 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bailleul, I., Gubinelli, M.: Unbounded rough drivers. ArXiv e-prints, (2015)Google Scholar
  5. 5.
    Bauzet, C.: Time-splitting approximation of the cauchy problem for a stochastic conservation law. Math. Comput. Simul. 118, 73–86 (2015)MathSciNetGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bauzet, C., Vallet, G., Wittbold, P.: The Cauchy problem for conservation laws with a multiplicative stochastic perturbation. J. Hyperbolic Differ. Equ. 9(4), 661–709 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bayer, C., Friz, P.K., Riedel, S., Schoenmakers, J.: From rough path estimates to multilevel monte carlo. SIAM J. Numer. Anal. 54(3), 1449–1483 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Biswas, I.H., Karlsen, K.H., Majee, A.K.: Conservation laws driven by Lévy white noise. J. Hyperbolic Differ. Equ. 12(3), 581–654 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen, G.-Q., Karlsen, K.H.: Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Commun. Pure Appl. Anal. 4(2), 241–266 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, G.-Q., Ding, Q., Karlsen, K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204(3), 707–743 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multilevel monte carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (2010)Google Scholar
  15. 15.
    Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259(4), 1014–1042 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Debussche, A., Vovelle, J.: Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Relat. Fields 163(3–4), 575–611 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Debussche, A., Hofmanová, M., Vovelle, J.: Degenerate parabolic stochastic partial differential equations: quasilinear case. Ann. Probab. 44(3), 1916–1955 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Deya, A., Gubinelli, M., Hofmanová, M., Tindel, S.: A priori estimates for rough PDEs with application to rough conservation laws. ArXiv e-prints, (2016)Google Scholar
  19. 19.
    Dotti, S., Vovelle, J.: Convergence of approximations to stochastic scalar conservation laws. ArXiv e-prints, (2016)Google Scholar
  20. 20.
    Dotti, S., Vovelle, J.: Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation. ArXiv e-prints, (2016)Google Scholar
  21. 21.
    Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1(1), 66–103 (1973)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Evans, L.C.: An Introduction to Stochastic Differential Equations. American Mathematical Society, Providence (2013)zbMATHGoogle Scholar
  23. 23.
    Feng, J., Nualart, D.: Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180(1), 1–53 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gaines, J.G., Lyons, T.J.: Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57(5), 1455–1484 (1997)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gassiat, P., Gess, B.: Regularization by noise for stochastic Hamilton-Jacobi equations. Probab. Theory Relat. Fields 173, 1–36 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gassiat, P., Gess, B., Lions, P.-L., Souganidis, P.E.: Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex hamiltonians. ArXiv e-prints, (2018)Google Scholar
  28. 28.
    Gess, B., Perthame, B., Souganidis, P.E.: Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes. SIAM J. Numer. Anal. 54(4), 2187–2209 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Gess, B., Souganidis, P.E.: Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Commun. Pure Appl. Math. 70(8), 1562–1597 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Gess, B., Souganidis, P.E.: Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. 13(6), 1569–1597 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Giles, M.B.: Multilevel monte carlo methods. Acta Numerica 24, 259–328 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Giles, M.B., Lester, C., Whittle, J.: Non-nested adaptive timesteps in multilevel Monte Carlo computations. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 303–314. Springer, Cham (2016)Google Scholar
  33. 33.
    Gut, A.: Probability: A Graduate Course, Springer Texts in Statistics, 2nd edn. Springer, New York (2013)zbMATHGoogle Scholar
  34. 34.
    Haji-Ali, A.-L., Nobile, F., Tempone, R.: Multi-index Monte Carlo: when sparsity meets sampling. Numerische Mathematik 132(4), 767–806 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Haji-Ali, A.-L., Nobile, F., von Schwerin, E., Tempone, R.: Optimization of mesh hierarchies in multilevel Monte Carlo samplers. Stoch. Partial Differ. Equ. Anal. Comput. 4(1), 76–112 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hall, E.J., Hoel, H., Sandberg, M., Szepessy, A., Tempone, R.: Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data. SIAM J. Sci. Comput. 38(6), A3773–A3807 (2016)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Hartmann, R., Houston, P.: Adaptive discontinuous galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24(3), 979–1004 (2003)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Hoel, H., Häppölä, J., Tempone, R.: Construction of a mean square error adaptive euler–maruyama method with applications in multilevel Monte Carlo. In: Cools, R., Nuyens, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 29–86. Springer, Cham (2016)Google Scholar
  39. 39.
    Hoel, H., Karlsen, K.H., Risebro, N.H., Storrøsten, E.B.: Path-dependent convex conservation laws. J. Differ. Equ. 265(6), 2708–2744 (2018)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hoel, H., Von Schwerin, E., Szepessy, A., Tempone, R.: Implementation and analysis of an adaptive multilevel Monte Carlo algorithm. Monte Carlo Methods Appl. 20(1), 1–41 (2014)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hofmanová, M.: Degenerate parabolic stochastic partial differential equations. Stoch. Process. Appl. 123(12), 4294–4336 (2013)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Hofmanová, M.: Scalar conservation laws with rough flux and stochastic forcing. Stoch. Partial Differ. Equ. Anal. Comput. 4(3), 635–690 (2016)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Holden, H., Risebro, N.H.: Conservation laws with a random source. Appl. Math. Optim. 36(2), 229–241 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws, Volume 152 of Applied Mathematical Sciences, 2nd edn. Springer, Heidelberg (2015)zbMATHGoogle Scholar
  45. 45.
    Johnson, C., Szepessy, A.: Adaptive finite element methods for conservation laws based on a posteriori error estimates. Commun. Pure Appl. Math. 48(3), 199–234 (1995)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, vol. 113. Springer, Berlin (1991)zbMATHGoogle Scholar
  47. 47.
    Karlsen, K.H., Risebro, N.H.: On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete Contin. Dyn. Syst. 9(5), 1081–1104 (2003)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Karlsen, K.H., Storrøsten, E.B.: On stochastic conservation laws and Malliavin calculus. J. Funct. Anal. 272, 421–497 (2017)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Karlsen, K.H., Storrøsten, E.B.: Analysis of a splitting method for stochastic balance laws. IMA J. Numer. Anal. 38(1), 1–56 (2018)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Kelly, C., Lord, G.J.: Adaptive time-stepping strategies for nonlinear stochastic systems. IMA J. Numer. Anal. 38, 1523–1549 (2016)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Kim, J.U.: On a stochastic scalar conservation law. Indiana Univ. Math. J. 52(1), 227–256 (2003)MathSciNetzbMATHGoogle Scholar
  52. 52.
    E, W., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877–960 (2000)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Kroese, D.P., Botev, Z.I.: Spatial Process Simulation. Stochastic Geometry. Spatial Statistics and Random Fields, pp. 369–404. Springer, Berlin (2015)zbMATHGoogle Scholar
  54. 54.
    Kröker, I., Rohde, C.: Finite volume schemes for hyperbolic balance laws with multiplicative noise. Appl. Numer. Math. 62(4), 441–456 (2012)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Kröner, D.: Numerical Schemes for Conservation Laws. Wiley-Teubner Series Advances in Numerical Mathematics. Wiley, Chichester (1997)zbMATHGoogle Scholar
  56. 56.
    Lions, P.-L., Perthame, B., Souganidis, P.: Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. Anal. Comput. 1(4), 664–686 (2013)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Lions, P.-L., Perthame, B., Souganidis, P.E.: Stochastic averaging lemmas for kinetic equations. ArXiv e-prints, (2012)Google Scholar
  58. 58.
    Lions, P.-L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case. Stoch. Partial Differ. Equ. Anal. Comput. 2(4), 517–538 (2014)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Lucier, B.J.: A moving mesh numerical method for hyperbolic conservation laws. Math. Comput. 46(173), 59–69 (1986)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Mishra, S., Schwab, C.: Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comput. 81(280), 1979–2018 (2012)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Mishra, S., Schwab, C., Šukys, J.: Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions. J. Comput. Phys. 231(8), 3365–3388 (2012)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Mohammed, S.-E.A., Nilssen, T.K., Proske, F.N.: Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation. Ann. Probab. 43(3), 1535–1576 (2015)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Mörters, P., Peres, Y.: Brownian motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  64. 64.
    Neves, W., Olivera, C.: Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1247–1258 (2015)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Perthame, B.: Kinetic Formulation of Conservation Laws Vol. 21. of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002)Google Scholar
  66. 66.
    Risebro, N.H., Schwab, C., Weber, F.: Multilevel Monte Carlo front-tracking for random scalar conservation laws. BIT Numer. Math. 56(1), 263–292 (2016)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Seeger, B.: Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations. ArXiv e-prints, (2018)Google Scholar
  68. 68.
    Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of stochastic differential equations. Commun. Pure Appl. Math. 54(10), 1169–1214 (2001)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Vallet, G.: Dirichlet problem for a nonlinear conservation law. Rev. Mat. Complut. 13(1), 231–250 (2000)MathSciNetzbMATHGoogle Scholar
  70. 70.
    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)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Yaroslavtseva, L.: On non-polynomial lower error bounds for adaptive strong approximation of sdes. J. Complex. 42, 1–18 (2017)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Mathematics Institute of Computational Science and EngineeringÉcole polytechnique fédérale de LausanneLausanneSwitzerland
  2. 2.Department of mathematicsUniversity of OsloOsloNorway

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