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High-order In-cell Discontinuous Reconstruction path-conservative methods for nonconservative hyperbolic systems -- DR.MOOD generalization (2407.02931v1)

Published 3 Jul 2024 in math.NA and cs.NA

Abstract: In this work we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in-cell discontinuous reconstruction operator are the key points to develop a new family of high-order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two-Layer Shallow Water system.

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References (31)
  1. R. Abgrall and S. Karni. A comment on the computation of non-conservative products. Journal of Computational Physics, 229(8):2759–2763, 2010.
  2. B. Audebert and F. Coquel. Hybrid Godunov-Glimm method for a nonconservative hyperbolic system with kinetic relations. In Numerical Mathematics and Advanced Applications, pages 646–653. Springer Berlin Heidelberg, 2006.
  3. Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form. Communications in Computational Physics, 21(4):913–946, 2017.
  4. C. Berthon. Schéma nonlinéaire pour l’approximation numérique d’un système hyperbolique non conservatif. Comptes Rendus Mathematique, 335(12):1069–1072, 2002.
  5. C. Berthon and F. Coquel. Nonlinear projection methods for multi-entropies Navier-Stokes systems. In Innovative Methods For Numerical Solution Of Partial Differential Equations, pages 278–304. World Scientific, 2002.
  6. B. Biswas and R. K. Dubey. Low dissipative entropy stable schemes using third order weno and tvd reconstructions. Advances in Computational Mathematics, 44:1153–1181, 2018.
  7. High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Mathematics of Computation, 75(255):1103–1135, 2006.
  8. A Q𝑄Qitalic_Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: Mathematical Modelling and Numerical Analysis, 35(1):107–127, 2001.
  9. Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems. SIAM Journal on Numerical Analysis, 51(3):1371–1391, 2013.
  10. Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. Journal of Computational Physics, 227(17):8107–8129, 2008.
  11. Well-Balanced Schemes and Path-Conservative Numerical Methods. In Handbook of Numerical Analysis, volume 18 of Handbook of Numerical Methods for Hyperbolic Problems Applied and Modern Issues, pages 131–175. Elsevier, 2017.
  12. C. Chalons. Path-conservative in-cell discontinuous reconstruction schemes for non conservative hyperbolic systems. Communications in Mathematical Sciences, 18(1):1–30, 2020.
  13. C. Chalons and F. Coquel. Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes. Numerische Mathematik, 101(3):451–478, 2005.
  14. C. Chalons and F. Coquel. A new comment on the computation of non-conservative products using Roe-type path conservative schemes. Journal of Computational Physics, 335:592–604, 2017.
  15. A high-order finite volume method for systems of conservation laws—multi-dimensional optimal order detection (mood). Journal of computational Physics, 230(10):4028–4050, 2011.
  16. Definition and weak stability of nonconservative products. Journal de mathématiques pures et appliquées, 74(6):483–548, 1995.
  17. B. Després and F. Lagoutière. Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. Journal of Scientific Computing, 16:479–524, 2001.
  18. Improved detection criteria for the multi-dimensional optimal order detection (mood) on unstructured meshes with very high-order polynomials. Computers & Fluids, 64:43–63, 2012.
  19. Ader schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows. Computers & Fluids, 38(9):1731–1748, 2009.
  20. U. S. Fjordholm and S. Mishra. Accurate numerical discretizations of non-conservative hyperbolic systems. ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 46(1):187–206, 2012.
  21. A. Harten and J. M. Hyman. Self adjusting grid methods for one-dimensional hyperbolic conservation laws. Journal of Computational Physics, 50(2):235–269, 1983.
  22. Entropy-stable space–time DG schemes for non-conservative hyperbolic systems. ESAIM: Mathematical Modelling and Numerical Analysis, 52(3):995–1022, 2018.
  23. F. Lagoutière. A non-dissipative entropic scheme for convex scalar equations via discontinuous cell-reconstruction. Comptes Rendus Mathematique, 338(7):549–554, 2004.
  24. F. Lagoutière. Stability of reconstruction schemes for scalar hyperbolic conservation laws. Commun. Math. Sci, 6(1):57–70, 2008.
  25. P. G. LeFloch and S. Mishra. Numerical methods with controlled dissipation for small-scale dependent shocks. Acta Numerica, 23:743–816, 2014.
  26. M. L. Muñoz-Ruiz and C. Parés. Godunov method for nonconservative hyperbolic systems. ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique, 41(1):169–185, 2007.
  27. C. Parés. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44(1):300–321, 2006.
  28. C. Parés and M. J. Castro-Díaz. On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: mathematical modelling and numerical analysis, 38(5):821–852, 2004.
  29. In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - second-order extension. Journal of Computational Physics, 459:111152, 2022.
  30. Ader schemes for three-dimensional non-linear hyperbolic systems. Journal of Computational Physics, 204(2):715–736, 2005.
  31. I. Toumi. A weak formulation of Roe’s approximate Riemann solver. Journal of Computational Physics, 102(2):360–373, 1992.
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