An alternative SPH formulation: ADER-WENO-SPH

Abstract We present a new class of fully-discrete one-step SPH schemes based on a mesh-free ADER (Arbitrary DERivatives in space and time) reconstruction on moving particles in multiple space dimensions. In particular, the new SPH scheme computes mesh-free and local high order accurate polynomials in space and time to evaluate numerical fluxes at the midpoint between two interaction particles with a proper Riemann solver within the general SPH framework of Vila (1999) for nonlinear systems of hyperbolic conservation laws. The new scheme has been carefully tested against reference solutions for both the compressible Euler and the magneto-hydrodynamics (MHD) equations. The capability of the proposed scheme to accurately capture shocks and rarefaction waves for 1D and 2D problems with minimal amount of diffusion has been demonstrated. Via numerical evidence it has been shown that the new fully-discrete one-step ADER-WENO-SPH method is computationally more efficient than WENO-SPH schemes based on classical Runge–Kutta time-stepping. This is mainly due to the fact that with ADER timestepping the expensive stencil and neighbor search needs to be done only once per time step, while with Runge–Kutta time integrators the neighbor and stencil search is needed in each Runge–Kutta stage again.

[1]  Salvatore Marrone,et al.  SPH accuracy improvement through the combination of a quasi-Lagrangian shifting transport velocity and consistent ALE formalisms , 2016, J. Comput. Phys..

[2]  David Le Touzé,et al.  An efficient FSI coupling strategy between Smoothed Particle Hydrodynamics and Finite Element methods , 2017, Comput. Phys. Commun..

[3]  Michael Dumbser,et al.  Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems , 2007, J. Comput. Phys..

[4]  Michael Dumbser,et al.  High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics , 2013, 1310.7256.

[5]  Michael Dumbser,et al.  A new class of Moving-Least-Squares WENO-SPH schemes , 2014, J. Comput. Phys..

[6]  Stéphane Clain,et al.  High-accurate SPH method with Multidimensional Optimal Order Detection limiting , 2016 .

[7]  Daniel J. Price,et al.  Constrained hyperbolic divergence cleaning in smoothed particle magnetohydrodynamics with variable cleaning speeds , 2016, J. Comput. Phys..

[8]  S. J. Lind,et al.  Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves , 2012, J. Comput. Phys..

[9]  G. Dilts.  MOVING-LEAST-SQUARES-PARTICLE HYDRODYNAMICS-I. CONSISTENCY AND STABILITY , 1999 .

[10]  Stéphane Clain,et al.  The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems , 2013 .

[11]  John S. Anagnostopoulos,et al.  An improved MUSCL treatment for the SPH‐ALE method: comparison with the standard SPH method for the jet impingement case , 2013 .

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

[13]  Michael Dumbser,et al.  Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes , 2013, 1302.3076.

[14]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[15]  E. J. Belin de Chantemèle,et al.  Long Term High Fat Diet Treatment: An Appropriate Approach to Study the Sex-Specificity of the Autonomic and Cardiovascular Responses to Obesity in Mice , 2017, Front. Physiol..

[16]  Stéphane Clain,et al.  Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials , 2012 .

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

[18]  Abouzied M. A. Nasar,et al.  High order difference schemes using the local anisotropic basis function method , 2019, J. Comput. Phys..

[19]  A. Colagrossi,et al.  δ-SPH model for simulating violent impact flows , 2011 .

[20]  N. Quinlan,et al.  Particle transport velocity correction for the finite volume particle method for multi-resolution particle distributions and exact geometric boundaries , 2020 .

[21]  Benedict D. Rogers,et al.  SPH for 3D floating bodies using variable mass particle distribution , 2013 .

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

[23]  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..

[24]  Eugenio Oñate,et al.  Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows , 2008 .

[25]  Eugenio Oñate,et al.  The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves , 2004 .

[26]  Eugenio Oñate,et al.  Multi-fluid flows with the Particle Finite Element Method , 2009 .

[27]  E. Oñate,et al.  Possibilities of the particle finite element method for fluid–soil–structure interaction problems , 2011 .

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

[29]  V. Springel.  E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh , 2009, 0901.4107.

[30]  Wing Kam Liu,et al.  Reproducing kernel particle methods for structural dynamics , 1995 .

[31]  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..

[32]  A. Gossler.  Moving Least-Squares: A Numerical Differentiation Method for Irregularly Spaced Calculation Points , 2001 .

[33]  S. Godunov,et al.  Elements of Continuum Mechanics and Conservation Laws , 2003, Springer US.

[34]  A. Bellin,et al.  Smooth Particle Hydrodynamics with nonlinear Moving-Least-Squares WENO reconstruction to model anisotropic dispersion in porous media , 2015 .

[35]  Guirong Liu,et al.  Smoothed Particle Hydrodynamics: A Meshfree Particle Method , 2003 .

[36]  Stefano Sibilla,et al.  An algorithm to improve consistency in Smoothed Particle Hydrodynamics , 2015 .

[37]  Steven J. Lind,et al.  High-order Eulerian incompressible smoothed particle hydrodynamics with transition to Lagrangian free-surface motion , 2016, J. Comput. Phys..

[38]  M. Dumbser,et al.  A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows , 2017 .

[39]  Salvatore Marrone,et al.  Free-surface flows solved by means of SPH schemes with numerical diffusive terms , 2010, Comput. Phys. Commun..

[40]  G. Russo,et al.  Central WENO schemes for hyperbolic systems of conservation laws , 1999 .

[41]  Benedict D. Rogers,et al.  Wave body interaction in 2D using smoothed particle hydrodynamics (SPH) with variable particle mass , 2012 .

[42]  Antonia Larese,et al.  Validation of the particle finite element method (PFEM) for simulation of free surface flows , 2008 .

[43]  Stéphane Clain,et al.  A high-order finite volume method for systems of conservation laws - Multi-dimensional Optimal Order Detection (MOOD) , 2011, J. Comput. Phys..

[44]  Stephen M. Longshaw,et al.  DualSPHysics: Open-source parallel CFD solver based on Smoothed Particle Hydrodynamics (SPH) , 2015, Comput. Phys. Commun..

[45]  D. Violeau,et al.  Calculating the smoothing error in SPH , 2019, Computers & Fluids.

[46]  Michael Dumbser,et al.  A new 3D parallel SPH scheme for free surface flows , 2009 .

[47]  Benedict D. Rogers,et al.  Simulation of caisson breakwater movement using 2-D SPH , 2010 .

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

[49]  J. K. Chen,et al.  A corrective smoothed particle method for boundary value problems in heat conduction , 1999 .

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

[51]  A. Bellin,et al.  An alternative smooth particle hydrodynamics formulation to simulate chemotaxis in porous media , 2016, Journal of mathematical biology.

[52]  Gabriella Puppo,et al.  Compact Central WENO Schemes for Multidimensional Conservation Laws , 1999, SIAM J. Sci. Comput..

[53]  Michael Dumbser,et al.  Central Weighted ENO Schemes for Hyperbolic Conservation Laws on Fixed and Moving Unstructured Meshes , 2017, SIAM J. Sci. Comput..

[54]  S. K. Godunov,et al.  Nonstationary equations of nonlinear elasticity theory in eulerian coordinates , 1972 .

[55]  M. Semplice,et al.  Adaptive Mesh Refinement for Hyperbolic Systems Based on Third-Order Compact WENO Reconstruction , 2014, Journal of Scientific Computing.

[56]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

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

[58]  J. Monaghan.  Simulating Free Surface Flows with SPH , 1994 .

[59]  L. Libersky,et al.  Smoothed Particle Hydrodynamics: Some recent improvements and applications , 1996 .

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

[61]  Joaquim Peiró,et al.  A smoothed particle hydrodynamics numerical scheme with a consistent diffusion term for the continuity equation , 2019, Computers & Fluids.

[62]  D. Balsara,et al.  A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations , 1999 .

[63]  Daniel Zwillinger,et al.  CRC standard mathematical tables and formulae; 30th edition , 1995 .

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

[65]  S. Orszag,et al.  Small-scale structure of two-dimensional magnetohydrodynamic turbulence , 1979, Journal of Fluid Mechanics.

[66]  Roland W. Lewis,et al.  A variational formulation based contact algorithm for rigid boundaries in two-dimensional SPH applications , 2004 .

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

[68]  Michael Dumbser,et al.  Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems , 2007, J. Comput. Phys..

[69]  P. Tesini,et al.  On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations , 2012, J. Comput. Phys..

[70]  Andrea Colagrossi,et al.  A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH , 2009, Comput. Phys. Commun..

[71]  Michael Dumbser,et al.  Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws , 2008, J. Comput. Phys..

[72]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[73]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

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

[75]  Gui-Rong Liu,et al.  Restoring particle consistency in smoothed particle hydrodynamics , 2006 .

[76]  Eugenio Oñate,et al.  The ALE/Lagrangian Particle Finite Element Method: A new approach to computation of free-surface flows and fluid–object interactions , 2007 .

[77]  Francis Leboeuf,et al.  Free surface flows simulations in Pelton turbines using an hybrid SPH-ALE method , 2010 .

[78]  Jean-Paul Vila,et al.  ON PARTICLE WEIGHTED METHODS AND SMOOTH PARTICLE HYDRODYNAMICS , 1999 .

[79]  Michael Dumbser,et al.  A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D , 2014, J. Comput. Phys..

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

[81]  José M. Domínguez,et al.  Local uniform stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models , 2019, Computers & Fluids.

[82]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[83]  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.

[84]  Michael Dumbser,et al.  Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations , 2009, J. Comput. Phys..

[85]  David Le Touzé,et al.  On distributed memory MPI-based parallelization of SPH codes in massive HPC context , 2016, Comput. Phys. Commun..

[86]  Rui Xu,et al.  Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach , 2009, J. Comput. Phys..

[87]  Salvatore Marrone,et al.  The δplus-SPH model: Simple procedures for a further improvement of the SPH scheme , 2017 .

[88]  Bachir Ben Moussa,et al.  On the Convergence of SPH Method for Scalar Conservation Laws with Boundary Conditions , 2006 .

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

[90]  Michael Dumbser,et al.  ADER-WENO finite volume schemes with space-time adaptive mesh refinement , 2012, J. Comput. Phys..

[91]  Gabriella Puppo,et al.  CWENO: Uniformly accurate reconstructions for balance laws , 2016, Math. Comput..