A high-order vertex-centered quasi-Lagrangian discontinuous Galerkin method for compressible Euler equations in two-dimensions

Abstract This article introduces a new vertex-centered quasi-Lagrangian discontinuous Galerkin method for two-dimensional compressible flows that is third-order accurate both in space and time. The computational domain is divided into structured quadrilateral cells with straight edges. Nodal control volumes with curved edges are constructed surrounding the grid vertices. The Euler equations in arbitrary Lagrangian-Eulerian (ALE) form are discretized on these nodal control volumes using a discontinuous Galerkin method. The time marching is implemented by the third-order strong-stability-preserving Runge-Kutta method. A compact HWENO reconstruction algorithm is used as limiter to eliminate spurious oscillations near discontinuities. The polynomial expression of fluid velocity defined in a nodal control volume is also obtained from the reconstruction procedure, and is used to calculate the moving velocity of corresponding grid vertex. In this way, the grid vertices are moved in a rigorously Lagrangian manner, although there are still mass fluxes between neighbouring nodal control volumes. Therefore the scheme is called as a quasi-Lagrangian one. The scheme is conservative for mass, momentum and total energy. Some numerical tests are carried out to demonstrate the accuracy and robustness of the scheme. It has a favorable qualitative behavior for discontinuous problems and optimal convergence rates for smooth problems.

[1]  Pierre-Henri Maire,et al.  A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes , 2009, J. Comput. Phys..

[2]  Michael Dumbser,et al.  Reprint of: Direct Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes , 2016, Computers & Fluids.

[3]  Antonio J. Gil,et al.  An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics , 2015, J. Comput. Phys..

[4]  Michael Dumbser,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[5]  Tzanio V. Kolev,et al.  High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics , 2012, SIAM J. Sci. Comput..

[6]  Pierre-Henri Maire,et al.  Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part II: The two-dimensional case , 2016, J. Comput. Phys..

[7]  Nathaniel R. Morgan,et al.  A Godunov-like point-centered essentially Lagrangian hydrodynamic approach , 2015, J. Comput. Phys..

[8]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[9]  Nathaniel R. Morgan,et al.  A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme , 2019, J. Comput. Phys..

[10]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[11]  George Em Karniadakis,et al.  Regular Article: A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains , 1999 .

[12]  Martin Berggren,et al.  A vertex-centered, dual discontinuous Galerkin method , 2006 .

[13]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[14]  Dimitri J. Mavriplis,et al.  On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes , 2008, J. Comput. Phys..

[15]  D. Benson.  Computational methods in Lagrangian and Eulerian hydrocodes , 1992 .

[16]  Chi-Wang Shu,et al.  A high order ENO conservative Lagrangian type scheme for the compressible Euler equations , 2007, J. Comput. Phys..

[17]  Mikhail J. Shashkov,et al.  One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian-Eulerian methods , 2012, J. Comput. Phys..

[18]  Juan Cheng,et al.  A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations , 2008 .

[19]  Michael Dumbser,et al.  High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes , 2019, J. Comput. Phys..

[20]  Nathaniel R. Morgan,et al.  A Lagrangian staggered grid Godunov-like approach for hydrodynamics , 2014, J. Comput. Phys..

[21]  L. Sedov.  Similarity and Dimensional Methods in Mechanics , 1960 .

[22]  Pavel B. Bochev,et al.  Fast optimization-based conservative remap of scalar fields through aggregate mass transfer , 2013, J. Comput. Phys..

[23]  Mark L. Wilkins,et al.  Use of artificial viscosity in multidimensional fluid dynamic calculations , 1980 .

[24]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[25]  Raphaël Loubère,et al.  Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme , 2011 .

[26]  Xiaodong Liu,et al.  Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates , 2018, J. Comput. Phys..

[27]  Gabriella Puppo,et al.  A Fourth-Order Central WENO Scheme for Multidimensional Hyperbolic Systems of Conservation Laws , 2002, SIAM J. Sci. Comput..

[28]  John K. Dukowicz,et al.  Vorticity errors in multidimensional Lagrangian codes , 1992 .

[29]  Elena Gaburro,et al.  A Unified Framework for the Solution of Hyperbolic PDE Systems Using High Order Direct Arbitrary-Lagrangian–Eulerian Schemes on Moving Unstructured Meshes with Topology Change , 2020 .

[30]  Chi-Wang Shu,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[31]  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.

[32]  Michael Dumbser,et al.  A second-order cell-centered Lagrangian ADER-MOOD finite volume scheme on multidimensional unstructured meshes for hydrodynamics , 2018, J. Comput. Phys..

[33]  Nathaniel R. Morgan,et al.  A Lagrangian discontinuous Galerkin hydrodynamic method , 2018 .

[34]  Jérôme Breil,et al.  Hybrid remap for multi-material ALE , 2011 .

[35]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[36]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[37]  J. Remacle,et al.  Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws , 2004 .

[38]  Carl Ollivier-Gooch,et al.  Accuracy preserving limiter for the high-order accurate solution of the Euler equations , 2009, J. Comput. Phys..

[39]  Mikhail Shashkov,et al.  An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics , 2019, Comput. Math. Appl..

[40]  Pierre-Henri Maire,et al.  Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case , 2015, J. Comput. Phys..

[41]  S. K. Godunov,et al.  Reminiscences about Difference Schemes , 1999 .

[42]  Michael Dumbser,et al.  Arbitrary-Lagrangian-Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes , 2016, J. Comput. Phys..

[43]  Vinh-Tan Nguyen,et al.  An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for simulations of flows over variable geometries , 2010 .

[44]  Rémi Abgrall,et al.  A Lagrangian Discontinuous Galerkin‐type method on unstructured meshes to solve hydrodynamics problems , 2004 .

[45]  Xijun Yu,et al.  The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two-dimensions , 2014 .

[46]  Chi-Wang Shu,et al.  Positivity-preserving Lagrangian scheme for multi-material compressible flow , 2014, J. Comput. Phys..

[47]  Shudao Zhang,et al.  A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions , 2011, J. Comput. Phys..

[48]  François Vilar,et al.  Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics , 2012 .

[49]  Ming Li,et al.  On the Geometric Conservation Law for Unsteady Flow Simulations on Moving Mesh , 2015 .

[50]  Rémi Abgrall,et al.  A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems , 2007, SIAM J. Sci. Comput..

[51]  R. Kidder,et al.  Laser-driven compression of hollow shells: power requirements and stability limitations , 1976 .

[52]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[53]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[54]  Zhi J. Wang,et al.  A Parameter-Free Generalized Moment Limiter for High- Order Methods on Unstructured Grids , 2009 .

[55]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[56]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[57]  William J. Rider,et al.  Arbitrary Lagrangian-Eulerian methods for modeling high-speed compressible multimaterial flows , 2016, J. Comput. Phys..

[58]  Philip L. Roe,et al.  A cell centred Lagrangian Godunov scheme for shock hydrodynamics , 2011 .

[59]  Jérôme Breil,et al.  Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods , 2011, J. Comput. Phys..

[60]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

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

[62]  Bruno Després,et al.  Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems , 2005 .

[63]  Nathaniel R. Morgan,et al.  A 3D finite element ALE method using an approximate Riemann solution , 2017 .

[64]  Marc R.J. Charest,et al.  A nodal Godunov method for Lagrangian shock hydrodynamics on unstructured tetrahedral grids , 2014 .

[65]  Rémi Abgrall,et al.  A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids , 2014, J. Comput. Phys..