An explicit residual based approach for shallow water flows

We describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids. The schemes are obtained by properly adapting the explicit construction proposed in Ricchiuto and Abgrall (2010) 57. In particular, previous work on well balanced integration (Ricchiuto, 2011 56) and preservation of the depth non-negativity (Ricchiuto and Bollermann, 2009 60) is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation. The paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria, and how to obtain a super-consistent approximation for smooth non-trivial moving equilibria. The treatment of the wetting/drying interface is also discussed, giving formal conditions for the preservation of the non-negativity of the depth for a particular case, based on a nonlinear variant of a Lax-Friedrichs type scheme. The approach is analyzed and tested thoroughly. The quality of the numerical results shows the interest in the proposed approach over previously proposed schemes, in terms of accuracy and efficiency.

[1]  Rémi Abgrall,et al.  A Lax–Wendroff type theorem for residual schemes , 2001 .

[2]  Carlos Parés,et al.  NUMERICAL TREATMENT OF WET/DRY FRONTS IN SHALLOW FLOWS WITH A MODIFIED ROE SCHEME , 2006 .

[3]  Pilar García-Navarro,et al.  A numerical model for the flooding and drying of irregular domains , 2002 .

[4]  Randall J. LeVeque,et al.  A Well-Balanced Path-Integral f-Wave Method for Hyperbolic Problems with Source Terms , 2011, J. Sci. Comput..

[5]  Nikolaos A. Kampanis,et al.  A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains , 2008 .

[6]  E. Tadmor.  Skew-selfadjoint form for systems of conservation laws , 1984 .

[7]  A. I. Delis,et al.  Relaxation schemes for the shallow water equations , 2003 .

[8]  Yulong Xing,et al.  On the Advantage of Well-Balanced Schemes for Moving-Water Equilibria of the Shallow Water Equations , 2011, J. Sci. Comput..

[9]  Pilar García-Navarro,et al.  Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique , 2003 .

[10]  Mario Ricchiuto,et al.  Stabilized residual distribution for shallow water simulations , 2009, J. Comput. Phys..

[11]  T. Gallouët,et al.  Some approximate Godunov schemes to compute shallow-water equations with topography , 2003 .

[12]  Mario Ricchiuto,et al.  Accuracy of stabilized residual distribution for shallow water flows including dry beds , 2008 .

[13]  P. G. Ciarlet,et al.  General lagrange and hermite interpolation in Rn with applications to finite element methods , 1972 .

[14]  Philip L. Roe,et al.  Fluctuations and signals - a framework for numerical evolution problems. , 1800 .

[15]  Rémi Abgrall,et al.  A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems: The July 2010 State of the Art , 2012 .

[16]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[18]  Yulong Xing,et al.  Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations , 2010 .

[19]  W. Thacker.  Some exact solutions to the nonlinear shallow-water wave equations , 1981, Journal of Fluid Mechanics.

[20]  M. Hubbard,et al.  Discontinuous upwind residual distribution: A route to unconditional positivity and high order accuracy , 2011 .

[21]  Mohammed Seaïd,et al.  Non‐oscillatory relaxation methods for the shallow‐water equations in one and two space dimensions , 2004 .

[22]  I. Nikolos,et al.  An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography , 2009 .

[23]  Chi-Wang Shu,et al.  On positivity preserving finite volume schemes for Euler equations , 1996 .

[24]  Valerio Caleffi,et al.  Finite volume method for simulating extreme flood events in natural channels , 2003 .

[25]  Rémi Abgrall,et al.  Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes , 2007, J. Comput. Phys..

[26]  Iñaki Garmendia,et al.  On the thermodynamics, stability and hierarchy of entropy functions in fluid flow , 2006 .

[27]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[28]  Michael J. Briggs,et al.  Laboratory experiments of tsunami runup on a circular island , 1995 .

[29]  Yulong Xing,et al.  High-order well-balanced finite volume WENO schemes for shallow water equation with moving water , 2007, J. Comput. Phys..

[30]  Manuel Jesús Castro Díaz,et al.  Available Online at Www.sciencedirect.com Mathematical and So,snos ~__d,~ot" Computer Modelling the Numerical Treatment of Wet/dry Fronts in Shallow Flows: Application to One-layer and Two-layer Systems , 2022 .

[31]  Philippe Bonneton,et al.  Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes , 2011 .

[32]  Yulong Xing,et al.  High order well-balanced schemes , 2010 .

[33]  Pietro Marco Congedo,et al.  Robust code-to-code comparison for long wave run up , 2013 .

[34]  Matthew E. Hubbard,et al.  A 2D numerical model of wave run-up and overtopping , 2002 .

[35]  Rémi Abgrall,et al.  Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems , 2003 .

[36]  Rémi Abgrall,et al.  Residual Distribution Schemes for Conservation Laws via Adaptive Quadrature , 2013, SIAM J. Sci. Comput..

[37]  Randall J. LeVeque,et al.  Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods , 1998 .

[38]  Mario Ricchiuto,et al.  On the C-property and Generalized C-property of Residual Distribution for the Shallow Water Equations , 2011, J. Sci. Comput..

[39]  Guillermo Hauke,et al.  A symmetric formulation for computing transient shallow water flows , 1998 .

[40]  Herman Deconinck,et al.  Residual distribution for general time-dependent conservation laws , 2005 .

[41]  Emmanuel Audusse,et al.  A well-balanced positivity preserving second-order scheme for shallow water flows on unstructured meshes , 2005 .

[42]  Yulong Xing,et al.  High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms , 2006, J. Comput. Phys..

[43]  F. Marche.  Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects , 2007 .

[44]  Numerical simulation of runoff over dry beds , 2003 .

[45]  Herman Deconinck,et al.  Residual Distribution Schemes: Foundations and Analysis , 2007 .

[46]  Luis Cea,et al.  Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations , 2012, J. Comput. Phys..

[47]  Pilar García-Navarro,et al.  Flux difference splitting and the balancing of source terms and flux gradients , 2000 .

[48]  A. Harten.  On the symmetric form of systems of conservation laws with entropy , 1983 .

[49]  C. E. Synolakis,et al.  Validation and Verification of Tsunami Numerical Models , 2008 .

[50]  S. Spekreijse.  Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws , 1986 .

[51]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[52]  Jostein R. Natvig,et al.  Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows , 2006, J. Comput. Phys..

[53]  Mario Ricchiuto,et al.  Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries , 2013, J. Comput. Phys..

[54]  Chi-Wang Shu,et al.  High-order finite volume WENO schemes for the shallow water equations with dry states , 2011 .

[55]  Mario Ricchiuto,et al.  Unconditionally stable space-time discontinuous residual distribution for shallow-water flows , 2013, J. Comput. Phys..

[56]  Rémi Abgrall,et al.  Residual distribution schemes: Current status and future trends , 2006 .

[57]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[58]  D. Birchall,et al.  Computational Fluid Dynamics , 2020, Radial Flow Turbocompressors.

[59]  Thomas J. R. Hughes,et al.  Stabilized Methods for Compressible Flows , 2010, J. Sci. Comput..

[60]  Philippe Bonneton,et al.  Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation , 2009 .

[61]  Alexandre Ern,et al.  A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying , 2008 .

[62]  Rémi Abgrall,et al.  Explicit Runge-Kutta residual distribution schemes for time dependent problems: Second order case , 2010, J. Comput. Phys..

[63]  Philip L. Roe,et al.  Computational fluid dynamics—retrospective and prospective , 2005 .

[64]  R. Abgrall,et al.  An Example of High Order Residual Distribution Scheme Using non-Lagrange Elements , 2010, J. Sci. Comput..

[65]  Pilar García-Navarro,et al.  Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography , 2004 .

[66]  Rémi Abgrall,et al.  Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems , 2006, J. Comput. Phys..

[67]  Harry Yeh,et al.  Advanced numerical models for simulating tsunami waves and runup , 2008 .

[68]  Joanna Szmelter,et al.  An edge-based unstructured mesh discretisation in geospherical framework , 2010, J. Comput. Phys..

[69]  John SantaLucia,et al.  THE THERMODYNAMICS OF , 2004 .

[70]  Joanna Szmelter,et al.  An edge-based unstructured mesh framework for atmospheric flows , 2011 .

[71]  Herman Deconinck,et al.  A Conservative Formulation of the Multidimensional Upwind Residual Distribution Schemes for General Nonlinear Conservation Laws , 2002 .

[72]  Tadmor.  Entropy functions for symmetric systems of conservation laws. Final Report , 1987 .

[73]  Rémi Abgrall,et al.  High Order Fluctuation Schemes on Triangular Meshes , 2003, J. Sci. Comput..

[74]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[75]  M. Ricchiuto.  Contributions to the development of residual discretizations for hyperbolic conservation laws with application to shallow water flows , 2011 .