Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

doi:10.1007/s42967-024-00369-y

Abstract

Abstract: In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical twopoint fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

Key words: Shallow water (SW) equations, Entropy conservation/dissipation, Uncertainty quantification, Discontinuous Galerkin (DG), Generalized Polynomial Chaos (gPC)

CLC Number:

Janina Bender, Philipp Öffner. Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1978-2010.

TrendMD

References

1. Abgrall, R.:A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. J. Comput. Phys. 372, 640-666 (2018). https://doi. org/10. 1016/j. jcp. 2018. 06. 031
2. Abgrall, R., Mishra, S.:Uncertainty qualification for hyperbolic systems of conservation laws. In: Handbook on Numerical Methods for Hyperbolic Problems. Applied and Modern Issues, pp. 507-544. Elsevier/North Holland, Amsterdam (2017). https://doi. org/10. 1016/bs. hna. 2016. 11. 003
3. Abgrall, R., Nordström, J., Öffner, P., Tokareva, S.:Analysis of the SBP-SAT stabilization for finite element methods. II:entropy stability. Commun. Appl. Math. Comput. 5(2), 573-595 (2023). https://doi. org/10. 1007/s42967- 020- 00086-2
4. Abgrall, R., Öffner, P., Ranocha, H.:Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes:application to structure preserving discretization. J. Comput. Phys. 453, 24 (2022). https://doi. org/10. 1016/j. jcp. 2022. 110955
5. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.:Julia:a fresh approach to numerical computing. SIAM Rev. 59(1), 65-98 (2017). https://doi. org/10. 1137/14100 0671
6. Cameron, R.H., Martin, W.T.:The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 2(48), 385-392 (1947). https://doi. org/10. 2307/19691 78
7. Chen, T., Shu, C.-W.:Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427-461 (2017). https://doi. org/10.1016/j. jcp. 2017. 05. 025
8. Chen, T., Shu, C.-W.:Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes. CSIAM Trans. Appl. Math. 1, 1-52 (2020)
9. Ciallella, M., Micalizzi, L., Öffner, P., Torlo, D.:An arbitrary high order and positivity preserving method for the shallow water equations. Comput. Fluids 247, 21 (2022). https://doi. org/10. 1016/j.compfluid. 2022. 105630
10. Ciallella, M., Torlo, D., Ricchiuto, M.:Arbitrary high order WENO finite volume scheme with flux globalization for moving equilibria preservation. J. Sci. Comput. 96(2), 28 (2023). https://doi. org/10. 1007/s10915- 023- 02280-9
11. Cockburn, B., Karniadakis, G.E., Shu, C.-W.:Discontinuous Galerkin Methods. Theory, Computation and Applications. In:Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)
12. Dai, D., Epshteyn, Y., Narayan, A.:Hyperbolicity-preserving and well-balanced stochastic Galerkin method for shallow water equations. SIAM J. Sci. Comput. 43(2), 929-952 (2021). https://doi. org/10. 1137/20M13 60736
13. Dai, D., Epshteyn, Y., Narayan, A.:Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations. J. Comput. Phys. 452, 28 (2022). https://doi. org/ 10. 1016/j. jcp. 2021. 110901
14. Després, B., Poëtte, G., Lucor, D.:Robust uncertainty propagation in systems of conservation laws with the entropy closure method. In:Uncertainty Quantification in Computational Fluid Dynamics, pp. 105-149. Springer, Heidelberg (2013). https://doi. org/10. 1007/978-3- 319- 00885-1_3
15. 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. 370, 22 (2020). https://doi. org/10. 1016/j. cam. 2019. 112602
16. Fisher, T.C., Carpenter, M.H., Nordström, J., Yamaleev, N.K., Swanson, C.:Discretely conservative finite-difference formulations for nonlinear conservation laws in split form:theory and boundary conditions. J. Comput. Phys. 234, 353-375 (2013). https://doi. org/10. 1016/j. jcp. 2012. 09. 026
17. Fjordholm, U.S., Mishra, S., Tadmor, E.:Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230(14), 5587-5609 (2011). https:// doi. org/10. 1016/j. jcp. 2011. 03. 042
18. Gaburro, E., Öffner, P., Ricchiuto, M., Torlo, D.:High order entropy preserving ADER-DG schemes. Appl. Math. Comput. 440, 21 (2023). https://doi. org/10. 1016/j. amc. 2022. 127644
19. Gassner, G.J.:A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), 1233-1253 (2013).https://doi. org/10. 1137/12089 0144
20. Gassner, G.J., Winters, A.R., Kopriva, D.A.:Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39-66 (2016). https://doi. org/10. 1016/j. jcp. 2016. 09. 013
21. Gassner, G.J., Winters, A.R., Kopriva, D.A.:A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291-308 (2016). https://doi. org/10. 1016/j. amc. 2015. 07. 014
22. Gerster, S., Herty, M.:Entropies and symmetrization of hyperbolic stochastic Galerkin formulations. Commun. Comput. Phys. 27(3), 639-671 (2020). https://doi. org/10. 4208/cicp. OA- 2019- 0047
23. Gerster, S., Herty, M., Sikstel, A.:Hyperbolic stochastic Galerkin formulation for the p-system. J. Comput. Phys. 395, 186-204 (2019). https://doi. org/10. 1016/j. jcp. 2019. 05. 049
24. Gerster, S., Sikstel, A., Visconti, G.:Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function. arXiv:2022- 03 (2022)
25. Gottlieb, D., Xiu, D.:Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3(2), 505-518 (2008)
26. Herty, M., Kolb, A., Müller, S.:Higher-Dimensional Deterministic Formulation of Hyperbolic Conservation Laws with Uncertain Initial Data. Institut für Geometrie und Praktische Mathematik, RWTH Aachen (2021)
27. Herty, M., Kolb, A., Müller, S.:Multiresolution analysis for stochastic hyperbolic conservation laws. IMA J. Numer. Anal. 44, 536-575 (2023). https://doi. org/10. 1093/imanum/drad0 10
28. Kuzmin, D., Hajduk, H., Rupp, A.:Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 389, 28 (2022). https://doi. org/10. 1016/j. cma. 2021. 114428
29. LeFloch, P.G., Mercier, J.M., Rohde, C.:Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968-1992 (2002). https://doi. org/10. 1137/S0036 14290 24006 9X
30. Mantri, Y., Öffner, P., Ricchiuto, M.:Fully well balanced entropy controlled DGSEM for shallow water flows:global flux quadrature and cell entropy correction. arXiv:2212. 11931 (2022)
31. Meister, A., Ortleb, S.:On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows. Int. J. Numer. Methods Fluids 76(2), 69-94 (2014). https://doi. org/ 10. 1002/fld. 3921
32. Michel, S., Torlo, D., Ricchiuto, M., Abgrall, R.:Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes:influence of approximation, stabilization, and time-stepping. J. Sci. Comput. 94, 49 (2023). https://doi. org/10. 1007/s10915- 022- 02087-0
33. Mishra, S., Risebro, N.H., Schwab, C., Tokareva, S.:Numerical solution of scalar conservation laws with random flux functions. SIAM/ASA J. Uncertain. Quantif. 4, 552-591 (2016). https://doi. org/10. 1137/12089 6967
34. Öffner, P.:Approximation and Stability Properties of Numerical Methods for Hyperbolic Conservation Laws. Springer, London (2023)
35. Öffner, P., Glaubitz, J., Ranocha, H.:Stability of correction procedure via reconstruction with summation-by-parts operators for Burgers' equation using a polynomial chaos approach. ESAIM Math. Model. Numer. Anal. 52(6), 2215-2245 (2018). https://doi. org/10. 1051/m2an/20180 72
36. Öffner, P., Ranocha, H., Sonar, T.:Correction procedure via reconstruction using summation-by-parts operators. In:Theory, Numerics and Applications of Hyperbolic Problems II, Aachen, Germany, August 2016, pp. 491-501. Springer, Cham (2018). https://doi. org/10. 1007/978-3- 319- 91548-7_37
37. Petrella, M., Tokareva, S., Toro, E.F.:Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries. J. Comput. Phys. 386, 405-427 (2019). https://doi. org/10. 1016/j. jcp. 2019. 02. 013
38. 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. In:Math. Eng. Springer, Cham (2015). https://doi. org/10. 1007/978-3- 319- 10714-1
39. Pettersson, P., Iaccarino, G., Nordström, J.:A stochastic Galerkin method for the Euler equations with Roe variable transformation. J. Comput. Phys. 257, 481-500 (2014). https://doi. org/10. 1016/j. jcp. 2013. 10. 011
40. Poëtte, G., Després, B., Lucor, D.:Uncertainty quantification for systems of conservation laws. J.Comput. Phys. 228(7), 2443-2467 (2009). https://doi. org/10. 1016/j. jcp. 2008. 12. 018
41. Ranocha, H.:Shallow water equations:split-form, entropy stable, well-balanced, and positivity preserving numerical methods. GEM. Int. J. Geomath. 8(1), 85-133 (2017). https://doi. org/10. 1007/ s13137- 016- 0089-9
42. Ranocha, H., Schlottke-Lakemper, M., Winters, A.R., Faulhaber, E., Chan, J., Gassner, G.:Adaptive numerical simulations with Trixi.jl:a case study of Julia for scientific computing. Proc. JuliaCon Conf. 1(1), 77 (2022). arXiv:2108. 06476
43. Ricchiuto, M., Abgrall, R., Deconinck, H.:Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. J. Comput. Phys. 222(1), 287- 331 (2007). https://doi. org/10. 1016/j. jcp. 2006. 06. 024
44. Schlachter, L., Schneider, F.:A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations. J. Comput. Phys. 375, 80-98 (2018). https://doi. org/10. 1016/j.jcp. 2018. 07. 026
45. Schlottke-Lakemper, M., Gassner, G.J., Ranocha, H., Winters, A.R., Chan, J.:Trixi.jl:adaptive high-order numerical simulations of hyperbolic PDEs in Julia. https://github. com/trixi- frame work/Trixi. jl (2021)
46. Schwab, C., Tokareva, S.:High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data. ESAIM Math. Model. Numer. Anal. 47(3), 807-835 (2013).https://doi. org/10. 1051/m2an/20120 60
47. Sonday, B.E., Berry, R.D., Najm, H.N., Debusschere, B.J.:Eigenvalues of the Jacobian of a Galerkinprojected uncertain ODE system. SIAM J. Sci. Comput. 33(3), 1212-1233 (2011). https://doi. org/10.1137/10078 5922
48. Tadmor, E.:The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49, 91-103 (1987). https://doi. org/10. 2307/20082 51
49. Tadmor, E.:Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451-512 (2003). https://doi. org/10. 1017/S0962 49290 20001 56
50. Tokareva, S., Schwab, C., Mishra, S.:High order SFV and mixed SDG/FV methods for the uncertainty quantification in multidimensional conservation laws. In:Lecture Notes in Computational Science and Engineering, vol. 99. Springer, Switzerland (2014). https://doi. org/10. 1007/978-3- 319- 05455-1_7
51. Wen, X., Don, W.S., Gao, Z., Xing, Y.:Entropy stable and well-balanced discontinuous Galerkin methods for the nonlinear shallow water equations. J. Sci. Comput. 83(3), 32 (2020). https://doi. org/10.1007/s10915- 020- 01248-3
52. Wiener, N.:The homogeneous chaos. Am. J. Math. 60, 897-936 (1938). https://doi. org/10. 2307/23712 68
53. Winters, A.R., Gassner, G.J.:A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography. J. Comput. Phys. 301, 357-376 (2015). https://doi. org/10. 1016/j. jcp. 2015. 08. 034
54. Wu, X., Kubatko, E.J., Chan, J.:High-order entropy stable discontinuous Galerkin methods for the shallow water equations:curved triangular meshes and GPU acceleration. Comput. Math. Appl. 82, 179-199 (2021). https://doi. org/10. 1016/j. camwa. 2020. 11. 006
55. Xiao, T., Kusch, J., Koellermeier, J., Frank, M.:A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws. J. Sci. Comput. (2023). https://doi. org/10. 1007/s10915- 023- 02143-3
56. Xing, Y., Shu, C.-W.:A survey of high order schemes for the shallow water equations. J. Math. Study 47(3), 221-249 (2014). https://doi. org/10. 4208/jms. v47n3. 14. 01
57. Xiu, D., Hesthaven, J.S.:High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118-1139 (2005). https://doi. org/10. 1137/04061 5201
58. Xiu, D., Karniadakis, G.E.:The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619-644 (2002). https://doi. org/10. 1137/S1064 82750 13878 26
59. Zhong, X., Shu, C.-W.:Entropy stable Galerkin methods with suitable quadrature rules for hyperbolic systems with random inputs. J. Sci. Comput. 92(1), 30 (2022). https://doi. org/10. 1007/s10915- 022- 01866-z

[1]	Joseph Hunter, Zheng Sun, Yulong Xing. Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 658-687.
[2]	Wei Guo, Jannatul Ferdous Ema, Jing-Mei Qiu. A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 550-575.
[3]	Bo Dong, Wei Wang. Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 311-324.
[4]	Mengjiao Jiao, Yan Jiang, Mengping Zhang. A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 279-310.
[5]	Fangyao Zhu, Juntao Huang, Yang Yang. Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 190-217.
[6]	Xiaofeng Cai, Wei Guo, Jing-Mei Qiu. Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 3-33.