345 research outputs found

    Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

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    This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

    Towards a new generation of multi-dimensional stellar evolution models: development of an implicit hydrodynamic code

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    This paper describes the first steps of development of a new multidimensional time implicit code devoted to the study of hydrodynamical processes in stellar interiors. The code solves the hydrodynamical equations in spherical geometry and is based on the finite volume method. Radiation transport is taken into account within the diffusion approximation. Realistic equation of state and opacities are implemented, allowing the study of a wide range of problems characteristic of stellar interiors. We describe in details the numerical method and various standard tests performed to validate the method. We present preliminary results devoted to the description of stellar convection. We first perform a local simulation of convection in the surface layers of a A-type star model. This simulation is used to test the ability of the code to address stellar conditions and to validate our results, since they can be compared to similar previous simulations based on explicit codes. We then present a global simulation of turbulent convective motions in a cold giant envelope, covering 80% in radius of the stellar structure. Although our implicit scheme is unconditionally stable, we show that in practice there is a limitation on the time step which prevent the flow to move over several cells during a time step. Nevertheless, in the cold giant model we reach a hydro CFL number of 100. We also show that we are able to address flows with a wide range of Mach numbers (10^-3 < Ms< 0.5), which is impossible with an anelastic approach. Our first developments are meant to demonstrate that the use of an implicit scheme applied to a stellar evolution context is perfectly thinkable and to provide useful guidelines to optimise the development of an implicit multi-D hydrodynamical code.Comment: 21 pages, 18 figures, accepted for publication in A&

    Meshless hydrodynamic simulations of young supernova remnants

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    The majority of massive stars end their lives by ejecting their outer envelopes in a corecollapse supernova explosion. The collision of their ejecta with the surrounding circumstellar medium results in the formation of supernova remnants that have been detected at all wavelengths, from radio to gamma-rays. For several dozen supernova remnants, very-long-baseline radio interferometers have spatially resolved the interaction region and directly measured the expansion rates of the shocked gas; many show evidence of the interaction of supernova ejecta with the dense slow winds characteristic of the red supergiant progenitors. Understanding the dynamics and morphology of the interaction region, particularly in young supernova remnants leads to estimates of the total mass of the circumstellar medium, as well as its density distribution around the star given the value of the wind velocity. Here we studied the interaction of the supernova ejecta with different circumstellar environments to investigate the hydrodynamic evolution of young supernova remnants in the SedovTaylor phase. We used the massively parallel, multi-physics magneto-hydrodynamics (MHD) and gravity code, GIZMO, for our simulations. We chose GIZMO for its flexibility in allowing the user to choose different methods to solve the fluid equations, i.e., new Lagrangian Godunovtype schemes, e.g., Meshless Finite Volume (MFV) and Meshless Finite Mass (MFM), as well as various flavors of smoothed particle hydrodynamics (SPH), or Eulerian fixed-grid schemes. Since the majority of previous studies used the latter, we focused on an extensive comparison of all the meshless methods in solving the Sedov-Taylor blastwave test, a problem for which there is an exact solution. For our given compute resources, we found the parameters (e.g., smoothing length, number of neighbours, artificial viscosity, and particle resolution) for each meshless method that gave the best agreement with the exact solution. We then carried out 2D and 3D simulations of the hydrodynamic interaction of the supernova ejecta with varying density profiles assumed for the circumstellar medium, namely: a 1/r 2 density profile, for a typical, spherically symmetric red supergiant stellar wind, and an axisymmetric torus profile, inspired by the observation of a dense, dusty torus of the circumstellar material around the red supergiant, WOH G64 (Ohnaka et al., 2008). Radially assembled Hierarchical Equal Area isoLatitude Pixelization (HEALPix) shells were used to set-up the initial density and velocity profiles for the ejecta, which is marked by a flat inner core and a steeply declining outer edge. The Weighted Voronoi Tessellation code was used to produce the 1/r 2 and axisymmetric torus density distributions. We showed that the growth of Richtmyer-Meshkov instabilities in the 2D and 3D 1/r 2 profiles are visible as early as 20 yrs into the evolution of the remnant and become increasingly unstable up to 100 yr. While 2D simulations of 1/r 2 profiles show the presence of the Richtmyer-Meshkov instabilities in the hot shell of a contact discontinuity, in 3D we see large bubbles and filamentary structure of the instabilities. Our results for the numerical approaches to simulating the systems for the 1/r 2 density cases were broadly consistent with previous studies in the literature where stationary grids were used. Two scenarios with different torus-cavity density contrasts were considered in which we found that the instability rolls along the half-opening angle takes ∼ 40 yr to develop in the axisymmetric torus with smooth density drop, whereas the axisymmetric torus with steep density drop does not develop any instability rolls up to the end of the simulation. We concluded with a discussion of the implications of our models for the morphology of supernova remnants and their expected levels of multi-wavelength emission

    Discontinuous Galerkin Spectral Element Methods for Astrophysical Flows in Multi-physics Applications

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    In engineering applications, discontinuous Galerkin methods (DG) have been proven to be a powerful and flexible class of high order methods for problems in computational fluid dynamics. However, the potential benefits of DG for applications in astrophysical contexts is still relatively unexplored in its entirety. To this day, a decent number of studies surveying DG for astrophysical flows have been conducted. But the adoption of DG by the astrophysics community is just beginning to gain traction and integration of DG into established, multi-physics simulation frameworks for comprehensive astrophysical modeling is still lacking. It is our firm believe, that the full potential of novel approaches for numerically solving the fluid equations only shows under the pressure of real-world simulations with all aspects of multi-physics, challenging flow configurations, resolution and runtime constraints, and efficiency metrics on high-performance systems involved. Thus, we see the pressing need to propel DG from the well-trodden path of cataloguing test results under "optimal" laboratory conditions towards the harsh and unforgiving environment of large-scale astrophysics simulations. Consequently, the core of this work is the development and deployment of a robust DG scheme solving the ideal magneto-hydrodynamics equations with multiple species on three-dimensional Cartesian grids with adaptive mesh refinement. We chose to implement DG within the venerable simulation framework FLASH, with a specific focus on multi-physics problems in astrophysics. This entails modifications of the vanilla DG scheme to make it fit seamlessly within FLASH in such a way that all other physics modules can be naturally coupled without additional implementation overhead. A key ingredient is that our DG scheme uses mean value data organized into blocks - the central data structure in FLASH. Having the opportunity to work on mean values, allows us to rely on a rock-solid, monotone Finite Volume (FV) scheme as "backup" whenever the high order DG method fails in cases when the flow gets too harsh. Finding ways to combine the two schemes in a fail-safe manner without loosing primary conservation while still maintaining high order accuracy for smooth, well-resolved flows involves a series of careful considerations, which we document in this thesis. The result of our work is a novel shock capturing scheme - a hybrid between FV and DG - with smooth transitions between low and high order fluxes according to solution smoothness estimators. We present extensive validations and test cases, specifically its interaction with multi-physics modules in FLASH such as (self-)gravity and radiative transfer. We also investigate the benefits and pitfalls of integrating end-to-end entropy stability into our numerical scheme, with special focus on highly compressible turbulent flows and shocks. Our implementation of DG in FLASH allows us to conduct preliminary yet comprehensive astrophysics simulations proving that our new solver is ready for assessments and investigations by the astrophysics community

    Numerical hydrodynamics in general relativity

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    The current status of numerical solutions for the equations of ideal general relativistic hydrodynamics is reviewed. With respect to an earlier version of the article the present update provides additional information on numerical schemes and extends the discussion of astrophysical simulations in general relativistic hydrodynamics. Different formulations of the equations are presented, with special mention of conservative and hyperbolic formulations well-adapted to advanced numerical methods. A large sample of available numerical schemes is discussed, paying particular attention to solution procedures based on schemes exploiting the characteristic structure of the equations through linearized Riemann solvers. A comprehensive summary of astrophysical simulations in strong gravitational fields is presented. These include gravitational collapse, accretion onto black holes and hydrodynamical evolutions of neutron stars. The material contained in these sections highlights the numerical challenges of various representative simulations. It also follows, to some extent, the chronological development of the field, concerning advances on the formulation of the gravitational field and hydrodynamic equations and the numerical methodology designed to solve them.Comment: 105 pages, 12 figures. The full online-readable version of this article, including several animations, will be published in Living Reviews in Relativity at http://www.livingreviews.or

    Entropy Viscosity Method for Lagrangian Hydrodynamics and Central Schemes for Mean Field Games

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    In this dissertation we consider two major subjects. The primary topic is the Entropy Viscosity method for Lagrangian hydrodynamics, the goal of which is to solve numerically the Euler equations of compressible gas dynamics. The second topic is concerned with applications of second order central differencing schemes to the Mean Field Games equations. The Entropy Viscosity method discretizes all kinematic and thermodynamic variables by high-order finite elements and solves the resulting discrete problem on a computational mesh that moves with the material velocity. The method is based on two major concepts. The first one is producing high order convergence rates for smooth solutions even with active viscosity terms. This is achieved by using high order finite element spaces and, more importantly, entropy based viscosity coefficients that clearly distinguish between smooth and singular regions. The second concept is providing control over oscillations around contact discontinuities as well as oscillations in shock regions. Achieving this requires adding extra viscosity terms in a way that the resulting system is still in agreement with generalized entropy inequalities, the minimum principle on the specific entropy and the general requirements for artificial tensor viscosities like orthogonal transformation invariance, radial symmetry, Galilean invariance, etc. We define a fully-discrete finite element algorithm and present numerical results on model Lagrangian hydro problems. We also discuss possible extensions of the method, e.g. length scale independent viscosity coefficients, incorporating mass diffusion into the mesh motion, and handling of different materials. In addition we present approaches to the different stages of arbitrary Lagrangian-Eulerian (ALE) methods, which can be used to extend the Entropy Viscosity method. That is, we discuss mesh relaxation by harmonic smoothing schemes, advection based solution remap, and multi-material zones treatment. The Mean Field Games (MFG) equations describe situations in which a large number of individual players choose their optimal strategy by considering global (but limited) incentive information that is available to everyone. The resulting system consists of a forward Hamilton-Jacobi equation and a backward convection-diffusion equation. We propose fully discrete explicit second order staggered finite difference schemes for the two equations and combine these schemes into a fixed point iteration algorithm. We discuss the second order accuracy of both schemes, their interaction in time, memory issues resulting from the forward-backward coupling, stopping criteria for the fixed point iteration, and parallel performance of the method
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