×

Entropies and symmetrization of hyperbolic stochastic Galerkin formulations. (English) Zbl 1518.65109

Summary: Stochastic quantities of interest are expanded in generalized polynomial chaos expansions using stochastic Galerkin methods. An application of hyperbolic differential equations in general does not transfer hyperbolicity to the coefficients of the truncated series expansion. For the Haar basis and for piecewise linear multiwavelets we present convex entropies for the systems of coefficients of the one-dimensional shallow water equations by using the Roe variable transform. This allows to obtain hyperbolicity, well-posedness and energy estimates.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35L65 Hyperbolic conservation laws
35R09 Integro-partial differential equations
54C70 Entropy in general topology
58J45 Hyperbolic equations on manifolds
37L45 Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35R60 PDEs with randomness, stochastic partial differential equations

Software:

ALEA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Abgrall, A simple, flexible and generic deterministic approach to uncertainty quantification in non-linear problems, Tech. report, 2008.
[2] R. Abgrall, P. Congedo, G. Geraci, and G. Iaccarino, An adaptive multiresolution semi-intrusive scheme for uq in compressible fluid problems, International Journal for Numerical Methods in Fluids 78 (2015), 595-637.
[3] R. Abgrall and S. Mishra, Uncertainty quantification for hyperbolic systems of conservation laws, Handbook of Numerical Analysis, vol. 18, pp. 507-544, Elsevier, 2017. · Zbl 1368.65205
[4] A. Bressan, Hyperbolic systems of conservation laws: The one dimensional Cauchy problem, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York, 2005. · Zbl 0997.35002
[5] R. Bürger and I. Kröker, Hybrid stochastic Galerkin finite volumes for the diffusively corrected Lighthill-Whitham-Richards traffic model, Finite Volumes for Complex Applications VIII -Hy-perbolic, Elliptic and Parabolic Problems (Cham), Springer International Publishing, 2017, pp. 189-197. · Zbl 1368.90045
[6] R. Bürger, I. Kröker, and C. Rohde, A hybrid stochastic Galerkin method for uncertainty quan-tification applied to a conservation law modelling a clarifier-thickener unit, ZAMM Journal of ap-plied mathematics and mechanics: Zeitschrift für angewandte Mathematik und Mechanik 94 (2014), no. 10, 793-817. · Zbl 1301.65126
[7] R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annals of Mathematics 48 (1947), no. 2, 385-392. · Zbl 0029.14302
[8] J. Carrillo, L. Pareschi, and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Communications in Computational Physics 25 (2019), 508-531. · Zbl 1473.65252
[9] G.-Q. Chen, Chapter 1 -Euler equations and related hyperbolic conservation laws, Handbook of Differential Equations: Evolutionary Equations, vol. 2, North-Holland, 2005, pp. 1-104. · Zbl 1092.35062
[10] Q.-Y. Chen, D. Gottlieb, and J. S. Hesthaven, Uncertainty analysis for the steady-state flows in a dual throat nozzle, Journal of Computational Physics 204 (2005), 378-398. · Zbl 1143.76430
[11] A. Chertock, S. Jin, and A. Kurganov, An operator splitting based stochastic Galerkin method for the one-dimensional compressible Euler equations with uncertainty, 2015.
[12] , A well-balanced operator splitting based stochastic Galerkin method for the one-dimensional Saint-Venant system with uncertainty, 2015.
[13] P. M. Congedo and M. Ricchiuto, Robust simulation of shallow water long wave runup, Congrès Français de Mécanique, 2013.
[14] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Wiley, New York, 1989. · Zbl 0729.35001
[15] I. Cravero, G. Puppo, M. Semplice, and G. Visconti, Cool WENO schemes, Computers & Fluids 169 (2018), 71-86. · Zbl 1410.76213
[16] , CWENO: Uniformly accurate reconstructions for balance laws, Mathematics of Compu-tation 87 (2018), 1689-1719. · Zbl 1412.65102
[17] R. Crisovan, D. Torlo, R. Abgrall, and S. Tokareva, Model order reduction for parametrized non-linear hyperbolic problems as an application to uncertainty quantification, Journal of Computa-tional and Applied Mathematics 348 (2019), 466-489. · Zbl 1404.65112
[18] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, 3 ed., A series of compre-hensive studies in mathematics, vol. 325, Springer-Verlag Berlin Heidelberg, 2010. · Zbl 1196.35001
[19] B. J. Debusschere, H. N Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem, and O. P. Le Maître, Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM Journal on Scientific Computing 26 (2004), no. 2, 698-719. · Zbl 1072.60042
[20] B. Després, G. Poëtte, and D. Lucor, Uncertainty quantification for systems of conservation laws, Journal of Computational Physics 228 (2009), 2443-2467. · Zbl 1161.65309
[21] , Robust uncertainty propagation in systems of conservation laws with the entropy clo-sure method, Uncertainty Quantification in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, vol. 92, Springer, Cham, 2013. · Zbl 1276.76072
[22] M. Eigel, C. J. Gittelson, C. Schwab, and E. Zander, Adaptive stochastic Galerkin FEM, Com-puter Methods in Applied Mechanics and Engineering 270 (2014), no. Supplement C, 247-269. · Zbl 1296.65157
[23] D. Funaro, Polynomial approximation of differential equations, vol. 8, Springer Science & Busi-ness Media, 2008. · Zbl 1170.78001
[24] S. Gerster, M. Herty, and A. Sikstel, Hyperbolic stochastic Galerkin formulation for the p-system, Journal of Computational Physics 395 (2019), 186-204. · Zbl 1452.65017
[25] R. G. Ghanem and P. D. Spanos, Stochastic finite elements: A spectral approach, 1 ed., Springer, New York, 1991. · Zbl 0722.73080
[26] J. Giesselmann, C. Makridakis, and T. Pryer, A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws, SIAM Journal on Numerical Analysis 53 (2014), 1280-1303. · Zbl 1457.65107
[27] J. Giesselmann, F. Meyer, and C. Rohde, A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method, IMA Journal of Numerical Analysis (2019). · Zbl 1464.65124
[28] E. Godlewski and P. A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, vol. 118, Applied Mathematical Sciences, Springer, New York, 1996. · Zbl 0860.65075
[29] S. K. Godunov and E. Romenskii, Elements of continuum mechanics and conservation laws, Springer US, 2003. · Zbl 1031.74004
[30] D. Gottlieb and J. S. Hesthaven, Spectral methods for hyperbolic problems, Journal of Computa-tional and Applied Mathematics 128 (2001), no. 1, 83-131. · Zbl 0974.65093
[31] D. Gottlieb and D. Xiu, Galerkin method for wave equations with uncertain coefficients, Commu-nications in computational physics 3 (2008), no. 2, 505-518. · Zbl 1195.65009
[32] M. D. Gunzburger, C. G. Webster, and G. Zhang, Stochastic finite element methods for partial differential equations with random input data, Acta Numerica 23 (2014), 521-650. · Zbl 1398.65299
[33] B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time-dependent problems and difference methods, 2 ed., Wiley, 2013. · Zbl 1275.65048
[34] A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen 69 (1910), 331-371. · JFM 41.0469.03
[35] J. Hadamard, Sur les problèmes aux dérivés partielles et leur signification physique, Princeton University Bulletin 13 (1902), 49-52.
[36] R. A. Horn and C. R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, New York, NY, USA, 2012.
[37] G. Hu, R. Li, and T. Tang, A robust WENO type finite volume solver for steady Euler equations on unstructured grids, Communications in Computational Physics 9 (2011), no. 3, 627-648. · Zbl 1364.65226
[38] J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, Journal of Computational Physics 315 (2016), 150-168. · Zbl 1349.82088
[39] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Com-putational Physics 26 (1996), no. 1, 202-228. · Zbl 0877.65065
[40] S. Jin and R. Shu, A study of hyperbolicity of kinetic stochastic Galerkin system for the isentropic Euler equations with uncertainty, Preprint (2018). · Zbl 1428.35212
[41] S. Jin, D. Xiu, and X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, Journal of Scientific Computing 67 (2016), 1198-1218. · Zbl 1342.65010
[42] S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales, SIAM Journal on Mathematical Analysis 50 (2018), no. 2, 1790-1816. · Zbl 1390.35295
[43] C. Klingenberg, G. Puppo, and M. Semplice, Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity, SIAM Journal on Scientific Computing 41 (2019), no. 2, 695-721. · Zbl 1412.65125
[44] I. Kröker, W. Nowak, and C. Rohde, A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems, Computational Geosciences 19 (2015), no. 2, 269-284. · Zbl 1396.65126
[45] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik 10 (1970), no. 2, 217. · Zbl 0215.16203
[46] J. Kusch and M. Frank, Intrusive methods in uncertainty quantification and their connection to kinetic theory, International Journal of Advances in Engineering Sciences and Applied Math-ematics 10 (2017), no. 1, 54-69. · Zbl 1397.65020
[47] C. D. Lellis, F. Otto, and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quarterly of Applied Mathematics 62 (2004), 687-700. · Zbl 1211.35184
[48] R. J. Leveque, Numerical methods for conservation laws, 2 ed., Lectures in Mathematics. ETH Zürich, Birkhäuser Basel, 1992. · Zbl 0847.65053
[49] , Finite volume methods for hyperbolic problems, 1 ed., Cambridge Texts in Applied Math-ematics, Cambridge University Press, 2002. · Zbl 1010.65040
[50] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineer-ing Sciences 229 (1955), no. 1178, 317-345. · Zbl 0064.20906
[51] T.-P. Liu, The Riemann problem for general 2×2 conservation laws, Transactions of the American Mathematical Society 199 (1974), 89-112. · Zbl 0289.35063
[52] , The Riemann problem for general systems of conservation laws, Journal of Differential Equations 18 (1975), no. 1, 218-234. · Zbl 0297.76057
[53] O. P. Le Maître and O. M. Knio, Spectral methods for uncertainty quantification, 1 ed., Springer Netherlands, 2010. · Zbl 1193.76003
[54] O. P. Le Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, Uncertainty propagation using Wiener-Haar expansions, Journal of Computational Physics 197 (2004), no. 1, 28-57. · Zbl 1052.65114
[55] S. Mishra and C. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyper-bolic conservation laws with random initial data, Mathematics of Computation 81 (2012), no. 280, 1979-2018. · Zbl 1271.65018
[56] S. Mishra, C. Schwab, and JŠukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, Journal of Computational Physics 231 (2012), no. 8, 3365-3388. · Zbl 1402.76083
[57] S. Mishra, C. Schwab, and J.Šukys, Multilevel Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions, SIAM Journal on Scientific Com-puting 34 (2012), no. 6, B761-B784. · Zbl 1263.76045
[58] M. Motamed, F. Nobile, and R. Tempone, A stochastic collocation method for the second order wave equation with a discontinuous random speed, Numerische Mathematik 123 (2013), no. 3, 493-536. · Zbl 1268.65008
[59] E. Y. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mathematical Notes 55 (1994), no. 5, 517-525. · Zbl 0839.35029
[60] P. Pettersson, G. Iaccarino, and J. Nordström, Numerical analysis of the Burgers’ equation in the presence of uncertainty, Journal of Computational Physics 228 (2009), no. 22, 8394-8412. · Zbl 1177.65017
[61] , A stochastic Galerkin method for the Euler equations with Roe variable transformation, Journal of Computational Physics 257 (2014), 481-500. · Zbl 1349.76251
[62] , Polynomial chaos methods for hyperbolic partial differential equations, Springer Interna-tional Publishing, Switzerland, 2015. · Zbl 1325.76004
[63] P. Pettersson and H. A. Tchelepi, Stochastic Galerkin framework with locally reduced bases for nonlinear two-phase transport in heterogeneous formations, Computer Methods in Applied Me-chanics and Engineering 310 (2016), 367-387. · Zbl 1439.65129
[64] R. Pulch and D. Xiu, Generalised polynomial chaos for a class of linear conservation laws, Journal of Scientific Computing 51 (2012), 293-312. · Zbl 1258.65008
[65] B. Riemann,Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abhand-lungen der Königlichen Gesellschaft der Wissenschaften in Göttingen 8 (1860), 43-66.
[66] N. H. Risebro, C. Schwab, and F. Weber, Multilevel Monte Carlo front-tracking for random scalar conservation laws, BIT Numerical Mathematics 56 (2016), no. 1, 263-292. · Zbl 1350.65008
[67] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics 43 (1981), 357-372. · Zbl 0474.65066
[68] C. Schwab and S. Tokareva, High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data, ESAIM: Mathematical Modelling and Numerical Analysis -Modélisation Mathématique et Analyse Numérique 47 (2013), no. 3, 807-835. · Zbl 1266.65008
[69] M. Semplice, A. Coco, and G. Russo, Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction, Journal of Scientific Computing 66 (2016), 692-724. · Zbl 1335.65077
[70] R. Shu, J. Hu, and S. Jin, A stochastic Galerkin method for the Boltzmann equation with multi-dimensional random inputs using sparse wavelet bases, Numerical Mathematics: Theory, Meth-ods and Applications 10 (2017), no. 2, 465-488. · Zbl 1399.65265
[71] J. Strikwerda, Finite difference schemes and partial differential equations, 2 ed., Society for Indus-trial and Applied Mathematics, 2004. · Zbl 1071.65118
[72] T. J. Sullivan, Introduction to uncertainty quantification, 1 ed., Texts in Applied Mathematics, Springer, Switzerland, 2015. · Zbl 1336.60002
[73] T. Tang and T. Zhou, Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed, Communications in Computational Physics 8 (2010), 226-248. · Zbl 1364.65215
[74] S. Tokareva, C. Schwab, and S. Mishra, High order SFV and mixed SDG/FV methods for the un-certainty quantification in multidimensional conservation laws, High Order Nonlinear Numerical Schemes for Evolutionary PDEs (Cham), Springer International Publishing, 2014, pp. 109-133. · Zbl 1421.76164
[75] S. Tokareva, C. Schwab, S. Mishra, and N. Risebro, Numerical solution of scalar conservation laws with random flux functions, SIAM/ASA Journal on Uncertainty Quantification 4 (2016), 552-591. · Zbl 1343.65007
[76] J. Tryoen, O. P. Le Maître, O. M. Knio, and A. Ern, Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws, SIAM Journal on Scientific Computing 34 (2012), no. 5, 2459-2481. · Zbl 1273.35330
[77] J. Tryoen, O. P. Le Maître, M. Ndjinga, and A. Ern, Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems, Journal of Computational Physics 229 (2010), 6485-6511. · Zbl 1197.65013
[78] J.Šukys, S. Mishra, and C. Schwab, Multi-level Monte Carlo finite difference and finite vol-ume methods for stochastic linear hyperbolic systems, Springer Proceedings in Mathematics and Statistics 65 (2013), 649-666. · Zbl 1302.65032
[79] N. Wiener, The homogeneous chaos, American Journal of Mathematics 60 (1938), no. 4, 897-936. · Zbl 0019.35406
[80] K. Wu, H. Tang, and D. Xiu, A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty, Journal of Computational Physics 345 (2017), 224-244. · Zbl 1380.65315
[81] D. Xiu, Numerical methods for stochastic computations, Princeton University Press, Princeton, 2010. · Zbl 1210.65002
[82] D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equa-tions, SIAM Journal on Scientific Computing 24 (2002), 619-644. · Zbl 1014.65004
[83] D. Xiu and J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, Journal of Computational Physics 228 (2009), no. 2, 266-281. · Zbl 1161.65008
[84] T. Zhou and T. Tang, Convergence analysis for spectral approximation to a scalar transport equation with a random wave speed, Journal of Computational Mathematics 30 (2012), no. 6, 643-656. · Zbl 1289.65012
[85] Y. Zhu and S. Jin, The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic preserving method, Multiscale Modeling & Simulation 15 (2017), 1502-1529. · Zbl 1378.82047
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.