On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

[1]  A. Treske,et al.  Undular bores (favre-waves) in open channels - Experimental studies , 1994 .

[2]  Jurjen A. Battjes,et al.  Experimental investigation of wave propagation over a bar , 1993 .

[3]  A. Berm'udez,et al.  A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows , 2020, J. Comput. Phys..

[4]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry , 1992 .

[5]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[6]  Jan S. Hesthaven,et al.  DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations , 2008 .

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

[8]  D. Bresch,et al.  On Navier–Stokes–Korteweg and Euler–Korteweg Systems: Application to Quantum Fluids Models , 2017, Archive for Rational Mechanics and Analysis.

[9]  Franklin Liu,et al.  Modeling wave runup with depth-integrated equations , 2002 .

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

[11]  Michael Dumbser,et al.  High Order ADER Schemes for Continuum Mechanics , 2020, Frontiers in Physics.

[12]  Eleuterio F. Toro,et al.  A projection hybrid high order finite volume/finite element method for incompressible turbulent flows , 2018, J. Comput. Phys..

[13]  Michael Dumbser,et al.  High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids , 2015, J. Comput. Phys..

[14]  Costas E. Synolakis,et al.  The runup of solitary waves , 1987, Journal of Fluid Mechanics.

[15]  Gianmarco Manzini,et al.  Mimetic finite difference method , 2014, J. Comput. Phys..

[16]  V. Duchêne.  Rigorous justification of the Favrie–Gavrilyuk approximation to the Serre–Green–Naghdi model , 2018, Nonlinearity.

[17]  Manuel Jesús Castro Díaz,et al.  High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems , 2009, J. Sci. Comput..

[18]  Michael Dumbser,et al.  A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems , 2011, J. Sci. Comput..

[19]  Gregor Gassner,et al.  Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations , 2017, J. Comput. Phys..

[20]  Manuel Torrilhon,et al.  On curl-preserving finite volume discretizations for shallow water equations , 2006 .

[21]  Michael Dumbser,et al.  A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies , 2020, Computers & Fluids.

[22]  Manuel Jesús Castro Díaz,et al.  Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws , 2008, SIAM J. Numer. Anal..

[23]  E. Madelung,et al.  Quantentheorie in hydrodynamischer Form , 1927 .

[24]  Eleuterio F. Toro,et al.  Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes , 2014, J. Comput. Phys..

[25]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. II: Unification of advection and diffusion , 2010, J. Comput. Phys..

[26]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

[27]  Sergey Gavrilyuk,et al.  Dispersive Nonlinear Waves in Two‐Layer Flows with Free Surface. I. Model Derivation and General Properties , 2007 .

[28]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[29]  I. Coddington,et al.  Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics , 2006 .

[30]  Michael Dumbser,et al.  High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension , 2020, J. Comput. Phys..

[31]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[32]  Jurjen A. Battjes,et al.  Numerical simulation of nonlinear wave propagation over a bar , 1994 .

[33]  P. M. Naghdi,et al.  A derivation of equations for wave propagation in water of variable depth , 1976, Journal of Fluid Mechanics.

[34]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[35]  P. Liu,et al.  A two-layer approach to wave modelling , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[36]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[37]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics , 1991 .

[38]  Philippe Bonneton,et al.  A fourth‐order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq‐type equations. Part I: model development and analysis , 2006 .

[39]  Michael Dumbser,et al.  An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes , 2019, J. Comput. Phys..

[40]  Cipriano Escalante,et al.  A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation , 2020, Journal of Scientific Computing.

[41]  Claus-Dieter Munz,et al.  Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model , 2000 .

[42]  Maurizio Brocchini,et al.  Dispersive nonlinear shallow-water equations: some preliminary numerical results , 2010 .

[43]  Manuel Jesús Castro Díaz,et al.  High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems , 2006, Math. Comput..

[44]  I. Müller,et al.  Rational Extended Thermodynamics , 1993 .

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

[46]  Michael Dumbser,et al.  Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations , 2010 .

[47]  C. Parés.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework. , 2006 .

[48]  Manuel Jesús Castro Díaz,et al.  Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme , 2017, Appl. Math. Comput..

[49]  Chris Kees,et al.  Robust explicit relaxation technique for solving the Green-Naghdi equations , 2019, J. Comput. Phys..

[50]  Doron Levy,et al.  Local discontinuous Galerkin methods for nonlinear dispersive equations , 2004 .

[51]  E. I. Romensky,et al.  Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics , 1998 .

[52]  Kwok Fai Cheung,et al.  Depth‐integrated, non‐hydrostatic model for wave breaking and run‐up , 2009 .

[53]  P. García-Navarro,et al.  On numerical treatment of the source terms in the shallow water equations , 2000 .

[54]  Michael Dumbser,et al.  Well-balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity , 2017, 1712.07765.

[55]  Dominique P. Renouard,et al.  Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle , 1987, Journal of Fluid Mechanics.

[56]  Michael Dumbser,et al.  A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws , 2014, J. Comput. Phys..

[57]  Sander Rhebergen,et al.  Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations , 2008, J. Comput. Phys..

[58]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[59]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[60]  E. Toro,et al.  Solution of the generalized Riemann problem for advection–reaction equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[61]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives , 2002, J. Sci. Comput..

[62]  M. Vázquez-Cendón.  Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

[63]  Ilya Peshkov,et al.  On a pure hyperbolic alternative to the Navier-Stokes equations , 2014 .

[64]  Eleuterio F. Toro,et al.  Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations , 2016, J. Comput. Phys..

[65]  Michael Dumbser,et al.  Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting , 2014, 1412.0081.

[66]  D. Bresch,et al.  A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations , 2016 .

[67]  Eleuterio F. Toro,et al.  Advection-Diffusion-Reaction Equations: Hyperbolization and High-Order ADER Discretizations , 2014, SIAM J. Sci. Comput..

[68]  Manuel Jesús Castro Díaz,et al.  On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas , 2007, J. Comput. Phys..

[69]  Alireza Mazaheri,et al.  A first-order hyperbolic system approach for dispersion , 2016, J. Comput. Phys..

[70]  M. J. Castro,et al.  ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows , 2009 .

[71]  A. Chesnokov,et al.  Hyperbolic model of internal solitary waves in a three-layer stratified fluid , 2020, The European Physical Journal Plus.

[72]  Eleuterio F. Toro,et al.  Derivative Riemann solvers for systems of conservation laws and ADER methods , 2006, J. Comput. Phys..

[73]  M. Shashkov,et al.  Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆ , 1997 .

[74]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..

[75]  Marcelo H. Kobayashi,et al.  Shock-capturing Boussinesq-type model for nearshore wave processes , 2010 .

[76]  Spencer J. Sherwin,et al.  An unstructured spectral/hp element model for enhanced Boussinesq-type equations , 2006 .

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

[78]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[79]  Spencer J. Sherwin,et al.  Spectral/hp discontinuous Galerkin methods for modelling 2D Boussinesq equations , 2006, J. Comput. Phys..

[80]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

[81]  Chi-Wang Shu,et al.  A Local Discontinuous Galerkin Method for KdV Type Equations , 2002, SIAM J. Numer. Anal..

[82]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[83]  Michael Dumbser,et al.  High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves , 2020, Applied Numerical Mathematics.

[84]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes , 2007, J. Comput. Phys..

[85]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

[86]  Michael Dumbser,et al.  Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems , 2019, Computers & Fluids.

[87]  Argiris I. Delis,et al.  Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations , 2014, J. Comput. Phys..

[88]  Carlos Parés,et al.  Godunov method for nonconservative hyperbolic systems , 2007 .

[89]  S. Gavrilyuk,et al.  A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves , 2017 .

[90]  Michael Dumbser,et al.  On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations , 2020, J. Comput. Phys..