New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

Abstract

In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.

Keywords

Hyperbolic conservation and balance laws with uncertainties / Finite-volume methods / Central-upwind schemes / Weighted essentially non-oscillatory (WENO)  interpolations

References

[1.] Abgrall R, Congedo PM. A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems. J. Comput. Phys., 2013, 235: 828-845

[2.] 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, vol. 18, pp. 507–544. Handbook of Numerical Analysis. Elsevier/North-Holland, Amsterdam (2017)

[3.] Abgrall, R., Tokareva, S.: The stochastic finite volume method. In: Shi, J., Lorenzo, P., (eds.) Uncertainty Quantification for Hyperbolic and Kinetic Equations, vol. 14, pp. 1–57. SEMA SIMAI Springer Series. Springer, Cham (2017)

[4.] Barth, T.: Non-intrusive uncertainty propagation with error bounds for conservation laws containing discontinuities. In: Hester, B., Didier, L., Siddhartha, M., Christoph, S. (eds.) Uncertainty Quantification in Computational Fluid Dynamics, vol. 92, pp. 1–57. Lecture Notes in Computer Science Engineering. Springer, Heidelberg (2013)

[5.] Bollermann A, Noelle S, Lukáčová-Medviďová M. Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys., 2011, 10: 371-404

[6.] Chorin AJ. Gaussian fields and random flow. J. Fluid Mech., 1974, 63: 21-32

[7.] Dai D, Epshteyn Y, Narayan A. Hyperbolicity-preserving and well-balanced stochastic Galerkin method for shallow water equations. SIAM J. Sci. Comput., 2021, 43: A929-A952

[8.] Dai D, Epshteyn Y, Narayan A. Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations. J. Comput. Phys., 2022, 452

[9.] Després, B., Poëtte, G., Lucor, D.: Robust uncertainty propagation in systems of conservation laws with the entropy closure method. In: Hester, B., Didier, L., Siddhartha, M., Christoph, S. (eds.) Uncertainty Quantification in Computational Fluid Dynamics. vol. 92, pp. 105–149. Lecture Notes in Computer Science Engineering. Springer, Heidelberg (2013)

[10.] Ditkowski A, Fibich G, Sagiv A. Density estimation in uncertainty propagation problems using a surrogate model. SIAM/ASA J. Uncertain. Quantif., 2020, 8: 261-300

[11.] Don WS, Li D-M, Gao Z, Wang B-S. A characteristic-wise alternative WENO-Z finite difference scheme for solving the compressible multicomponent non-reactive flows in the overestimated quasi-conservative form. J. Sci. Comput., 2020, 82: 27

[12.] Don WS, Li R, Wang B-S, Wang YH. A novel and robust scale-invariant WENO scheme for hyperbolic conservation laws. J. Comput. Phys., 2022, 448

[13.] Dürrwächter J, Kuhn T, Meyer F, Schlachter L, Schneider F. A hyperbolicity-preserving discontinuous stochastic Galerkin scheme for uncertain hyperbolic systems of equations. J. Comput. Appl. Math., 2020, 370

[14.] Geraci G, Congedo PM, Abgrall R, Iaccarino G. A novel weakly-intrusive non-linear multiresolution framework for uncertainty quantification in hyperbolic partial differential equations. J. Sci. Comput., 2016, 66: 358-405

[15.] Gerster S, Herty M. Entropies and symmetrization of hyperbolic stochastic Galerkin formulations. Commun. Comput. Phys., 2020, 27: 639-671

[16.] Gerster S, Herty M, Iacomini E. Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation. Math. Biosci. Eng., 2021, 18: 4372-4389

[17.] Gerster S, Herty M, Sikstel A. Hyperbolic stochastic Galerkin formulation for the p \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-system. J. Comput. Phys., 2019, 395: 186-204

[18.] Gottlieb, S., Ketcheson, D., Shu, C.-W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific Publishing Co. Pte. Ltd, Hackensack (2011)

[19.] Gottlieb S, Shu C-W, Tadmor E. Strong stability-preserving high-order time discretization methods. SIAM Rev., 2001, 43: 89-112

[20.] Jakeman J, Archibald R, Xiu D. Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids. J. Comput. Phys., 2011, 230: 3977-3997

[21.] Jiang, Y., Shu, C.-W., Zhang, M.: An alternative formulation of finite difference weighted ENO schemes with Lax-Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35, A1137–A1160 (2013)

[22.] Jin S, Shu R. A study of hyperbolicity of kinetic stochastic Galerkin system for the isentropic Euler equations with uncertainty. Chin. Ann. Math. Ser. B, 2019, 40: 765-780

[23.] Kurganov, A., Noelle, S., Petrova, G.: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)

[24.] Kurganov A, Petrova G. A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci., 2007, 5: 133-160

[25.] Kurganov, A., Tadmor, E.: Solution of two-dimensional riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differential Equations 18, 584–608 (2002)

[26.] Le Maître, O. P., Knio, O. M.: Spectral Methods for Uncertainty Quantification. Springer, New York (2010)

[27.] Le Maître, O.P., Knio, O.M., Najm, H.N., Ghanem, R.G.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197, 28–57 (2004)

[28.] Li P, Li TT, Don WS, Wang B-S. Scale-invariant multi-resolution alternative WENO scheme for the Euler equations. J. Sci. Comput., 2023, 94: 15

[29.] Lie K-A, Noelle S. On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput., 2003, 24: 1157-1174

[30.] Liu H. A numerical study of the performance of alternative weighted ENO methods based on various numerical fluxes for conservation law. Appl. Math. Comput., 2017, 296: 182-197

[31.] Mishra S, Schwab C. Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp., 2012, 81: 1979-2018

[32.] Mishra, S., Schwab, C.: Monte-Carlo finite-volume methods in uncertainty quantification for hyperbolic conservation laws. In: Shi, J., Lorenzo, P. (eds.) Uncertainty Quantification for Hyperbolic and Kinetic Equations, vol. 14, pp. 231–277. SEMA SIMAI Springer Series. Springer, Cham (2017)

[33.] 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., 2012, 231: 3365-3388

[34.] Mishra S, Schwab C, Šukys J. Multilevel Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions. SIAM J. Sci. Comput., 2012, 34: B761-B784

[35.] Mishra, S., Schwab, C., Šukys, J.: Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws. In: Hester, B., Didier, L., Siddhartha, M., Christoph, S. (eds.) Uncertainty Quantification in Computational Fluid Dynamics, vol. 92, pp. 225–294. Lecture Notes in Engineering and Computer Science Engineering. Springer, Heidelberg (2013)

[36.] Nessyahu H, Tadmor E. Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 1990, 87: 408-463

[37.] Petrella M, Tokareva S, Toro EF. Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries. J. Comput. Phys., 2019, 386: 405-427

[38.] Pettersson, M.P., Iaccarino, G., Nordström, J.: A stochastic Galerkin method for the Euler equations with Roe variable transformation. J. Comput. Phys. 257, 481–500 (2014)

[39.] Pettersson, M.P., Iaccarino, G., Nordström, J.: Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Springer, Cham (2015)

[40.] Poëtte G, Després B, Lucor D. Uncertainty quantification for systems of conservation laws. J. Comput. Phys., 2009, 228: 2443-2467

[41.] Schlachter L, Schneider F. A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations. J. Comput. Phys., 2018, 375: 80-98

[42.] Shi J, Hu C, Shu C-W. A technique of treating negative weights in WENO schemes. J. Comput. Phys., 2002, 175: 108-127

[43.] Šukys, J., Mishra, S., Schwab, C.: Multi-level Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems. In: Josef, D., Frances, Y. K., Gareth, W. P., Ian, H. S. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2012, vol. 65, pp. 649–666. Springer Proceedings in Mathematics and Statistics. Springer, Heidelberg (2013)

[44.] Sweby PK. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal., 1984, 21: 995-1011

[45.] Tang T, Zhou T. Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed. Commun. Comput. Phys., 2010, 8: 226-248

[46.] Tokareva S, Zlotnik A, Gyrya V. Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks. Appl. Math. Model., 2024, 125: 66-84

[47.] Tryoen J, Le Maître O, Ndjinga M, Ern A. Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys., 2010, 229: 6485-6511

[48.] Tryoen J, Le Maitre O, Ndjinga M, Ern A. Roe solver with entropy corrector for uncertain hyperbolic systems. J. Comput. Appl. Math., 2010, 235: 491-506

[49.] Wan X, Karniadakis GE. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput., 2006, 28: 901-928

[50.] Wang B-S, Don WS. Affine-invariant WENO weights and operator. Appl. Numer. Math., 2022, 181: 630-646

[51.] Wang B-S, Li P, Gao Z, Don WS. An improved fifth order alternative WENO-Z finite difference scheme for hyperbolic conservation laws. J. Comput. Phys., 2018, 374: 469-477

[52.] Wu K, Tang H, Xiu D. A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty. J. Comput. Phys., 2017, 345: 224-244

[53.] Wu K, Xiu D, Zhong X. A WENO-based stochastic Galerkin scheme for ideal MHD equations with random inputs. Commun. Comput. Phys., 2021, 30: 423-447

[54.] Xiu D. Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys., 2007, 2: 293-309

[55.] Xiu D. Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys., 2009, 5: 242-272

[56.] Xiu, D.: Numerical Methods for Stochastic Computations. A Spectral Method Approach. Princeton University Press, Princeton (2010)

[57.] Zhong X, Shu C-W. Entropy stable Galerkin methods with suitable quadrature rules for hyperbolic systems with random inputs. J. Sci. Comput., 2022, 92: 14

Funding

Division of Mathematical Sciences(DMS-2208438); Deutsche Forschungsgemeinschaft(HE5386/ 18-1, 19-2, 22-1, 23-1); Germany’s Excellence Strategy EXC-2023 Internet of Production(390621612); National Natural Science Foundation of China(12171226); Guangdong Provincial Key Laboratory Of Computational Science And Material Design(2019B030301001); Deutsche Forschungsgemeinschaft(525853336 funded within the Focused Programme SPP 2410 “Hypebrolic Balance Laws: Complexity, Scales and Randomness”); LeRoy B. Martin, Jr. Distinguished Professorship Foundation; LeRoy B. Martin, Jr. Distinguished Professorship Foundation