Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows

An approach for constructing a low-dissipation numerical method is described. The method is based on a combination of the operator-splitting method, Godunov method, and piecewise-parabolic method on the local stencil. Numerical method was tested on a standard suite of hydrodynamic test problems. In addition, the performance of the method is demonstrated on a global test problem showing the development of a spiral structure in a gravitationally unstable gaseous galactic disk. New high order accurate numerical method.New limiter-free, Galilean invariance, guaranteed non-decrease of the entropy numerical method.Numerical simulation of 2-4-7-arms spiral structure.

[1]  Boltzmann moment equation approach for the numerical study of anisotropic stellar discs , 2006, astro-ph/0609250.

[2]  Michael Dumbser,et al.  Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics , 2008, Journal of Computational Physics.

[3]  James Wadsley,et al.  On the treatment of entropy mixing in numerical cosmology , 2008 .

[4]  Igor Kulikov.  PEGAS: Hydrodynamical code for numerical simulation of the gas components of interacting galaxies , 2013 .

[5]  Michael Dumbser,et al.  Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers , 2013, J. Comput. Phys..

[6]  James Binney,et al.  Galactic Dynamics: Second Edition , 2008 .

[7]  Rémi Abgrall,et al.  Multidimensional HLLC Riemann solver for unstructured meshes - With application to Euler and MHD flows , 2014, J. Comput. Phys..

[8]  D. Lynden-Bell,et al.  II. Spiral Arms as Sheared Gravitational Instabilities , 1965 .

[9]  A. Whitworth,et al.  Adaptive smoothing lengths in SPH , 2007, astro-ph/0701909.

[10]  A. Toomre,et al.  On the gravitational stability of a disk of stars , 1964 .

[11]  S. K. Godunov,et al.  Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee , 2014 .

[12]  E. Athanassoula,et al.  The spiral structure of galaxies , 1984 .

[13]  Michael Dumbser,et al.  Multidimensional Riemann problem with self-similar internal structure. Part II - Application to hyperbolic conservation laws on unstructured meshes , 2015, J. Comput. Phys..

[14]  D. A. Clarke.  ON THE RELIABILITY OF ZEUS-3D , 2010 .

[15]  I. Kulikov,et al.  Gas dynamics of a central collision of two galaxies: Merger, disruption, passage, and the formation of a new galaxy , 2011 .

[16]  J. Monaghan,et al.  Fundamental differences between SPH and grid methods , 2006, astro-ph/0610051.

[17]  Igor G. Chernykh,et al.  AstroPhi: A code for complex simulation of the dynamics of astrophysical objects using hybrid supercomputers , 2015, Comput. Phys. Commun..

[18]  E. Tadmor,et al.  New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations , 2000 .

[19]  Chongam Kim,et al.  Multi-dimensional limiting process for three-dimensional flow physics analyses , 2008, J. Comput. Phys..

[20]  Russia,et al.  Collisionless stellar hydrodynamics as an efficient alternative to N-body methods , 2012, 1210.5246.

[21]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[22]  S. Godunov,et al.  Numerical and experimental simulation of wave formation during explosion welding , 2013, Proceedings of the Steklov Institute of Mathematics.

[23]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[24]  V. A. Vshivkov,et al.  HYDRODYNAMICAL CODE FOR NUMERICAL SIMULATION OF THE GAS COMPONENTS OF COLLIDING GALAXIES , 2011 .

[25]  H. Deconinck,et al.  Design principles for bounded higher-order convection schemes - a unified approach , 2007, J. Comput. Phys..

[26]  Igor M. Kulikov,et al.  Supercomputer Simulation of an Astrophysical Object Collapse by the Fluids-in-Cell Method , 2009, PaCT.

[27]  M. V. Popov,et al.  Piecewise parabolic method on local stencil for gasdynamic simulations , 2007 .

[28]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[29]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[30]  V. Springel,et al.  Cosmological smoothed particle hydrodynamics simulations: the entropy equation , 2001, astro-ph/0111016.

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

[32]  Devin W. Silvia,et al.  ENZO: AN ADAPTIVE MESH REFINEMENT CODE FOR ASTROPHYSICS , 2013, J. Open Source Softw..

[33]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

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

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

[36]  M. V. Popov,et al.  Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics , 2008 .

[37]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[38]  Dynamics of Circumstellar Disks , 1998, astro-ph/9802191.

[39]  Igor Kulikov,et al.  GPUPEGAS: A NEW GPU-ACCELERATED HYDRODYNAMIC CODE FOR NUMERICAL SIMULATIONS OF INTERACTING GALAXIES , 2014 .

[40]  Austria,et al.  The effect of external environment on the evolution of protostellar disks , 2014, 1410.1743.

[41]  Dinshaw S. Balsara.  Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows , 2010, J. Comput. Phys..

[42]  Dinshaw S. Balsara,et al.  Maintaining Pressure Positivity in Magnetohydrodynamic Simulations , 1999 .

[44]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[45]  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.

[46]  A modified fluids-in-cell method for problems of gravitational gas dynamics , 2007 .

[47]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[48]  Dinshaw S. Balsara,et al.  Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics , 2012, J. Comput. Phys..

[49]  Michael Dumbser,et al.  Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers , 2015, J. Comput. Phys..

[50]  Princeton University,et al.  BETHE-Hydro: An Arbitrary Lagrangian-Eulerian Multidimensional Hydrodynamics Code for Astrophysical Simulations , 2008, 0805.3356.

[51]  Austria,et al.  Self-gravitating equilibrium models of dwarf galaxies and the minimum mass for star formation , 2012, 1205.1537.

[52]  Dinshaw S. Balsara,et al.  Three dimensional HLL Riemann solver for conservation laws on structured meshes; Application to Euler and magnetohydrodynamic flows , 2015, J. Comput. Phys..

[53]  Dinshaw S. Balsara.  A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows , 2012, J. Comput. Phys..

[54]  V. Springel,et al.  Physical viscosity in smoothed particle hydrodynamics simulations of galaxy clusters , 2006, astro-ph/0605301.

[55]  Alexei V. Snytnikov,et al.  Computational methods for ill-posed problems of gravitational gasodynamics , 2011 .

[56]  Dinshaw S. Balsara,et al.  Multidimensional Riemann problem with self-similar internal structure. Part I - Application to hyperbolic conservation laws on structured meshes , 2014, J. Comput. Phys..

[57]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[58]  Linear Differential Relations Between Solutions of the Class of Euler-Poisson-Darboux Equations , 2005 .

[59]  Jeremiah P. Ostriker,et al.  A Cosmological Hydrodynamic Code Based on the Total Variation Diminishing Scheme , 1993 .

[60]  Canada.,et al.  Shape and orientation of stellar velocity ellipsoids in spiral galaxies , 2007, 0709.2768.

[61]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .

[62]  E. Tasker,et al.  A test suite for quantitative comparison of hydrodynamic codes in astrophysics , 2008, 0808.1844.

[63]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .