Factorized Runge-Kutta-Chebyshev Methods [CL]


The second-order extended stability Factorized Runge-Kutta-Chebyshev (FRKC2) class of explicit schemes for the integration of large systems of PDEs with diffusive terms is presented. FRKC2 schemes are straightforward to implement through ordered sequences of forward Euler steps with complex stepsizes, and easily parallelised for large scale problems on distributed architectures.
Preserving 7 digits for accuracy at 16 digit precision, the schemes are theoretically capable of maintaining internal stability at acceleration factors in excess of 6000 with respect to standard explicit Runge-Kutta methods. The stability domains have approximately the same extents as those of RKC schemes, and are a third longer than those of RKL2 schemes. Extension of FRKC methods to fourth-order, by both complex splitting and Butcher composition techniques, is discussed.
A publicly available implementation of the FRKC2 class of schemes may be obtained from maths.dit.ie/frkc

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S. OSullivan
Thu, 16 Feb 17

Comments: 9 pages, 6 figures, accepted to the proceedings of Astronum 2016 – 11th Annual International Conference on Numerical Modeling of Space Plasma Flows, June 6-10, 2016

A numerical scheme for the compressible low-Mach number regime of ideal fluid dynamics [CL]


Based on the Roe solver a new technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations was proposed in Miczek et al.: New numerical solver for flows at various mach numbers. A&A 576, A50 (2015). We analyze properties of this scheme and demonstrate that its limit yields a discretization of the continuous limit system. Furthermore we perform a linear stability analysis for the case of explicit time integration and study the performance of the scheme under implicit time integration via the evolution of its condition number. A numerical implementation demonstrates the capabilities of the scheme on the example of the Gresho vortex which can be accurately followed down to Mach numbers of ~1e-10 .

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W. Barsukow, P. Edelmann, C. Klingenberg, et. al.
Wed, 14 Dec 16

Comments: N/A

Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations [CL]


This paper presents an arbitrary h.o. accurate ADER DG method on space-time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent PDE for compr. dissipative flows: the compr. Navier-Stokes equations and the equations of visc. and res. MHD in 2 and 3 space-dimensions. The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell FV limiting procedure suitable for DG methods. It is a well known fact that a major weakness of h.o. DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ‘Gibbs phenomenon’. In the present work the main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. An important feature of our new scheme is its ability to cure even floating point errors that may occur during a simulation, for example when taking real roots of negative numbers or after divisions by zero. We apply the whole approach for the first time to the equations of compr. gas dynamics and MHD in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector. The distinguished high-resolution properties of the presented numerical scheme stands out against a wide number of non-trivial test cases both for the compr. Navier-Stokes and the viscous and resistive MHD equations. The present results show clearly that the shock-capturing capability of the news schemes are significantly enhanced within a cell-by-cell Adaptive Mesh Refinement implementation together with time accurate local time stepping (LTS).

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F. Fambri, M. Dumbser and O. Zanotti
Tue, 6 Dec 16

Comments: 31 pages, 16 figures

Scaling Laws of Passive-Scalar Diffusion in the Interstellar Medium [GA]


Passive scalar mixing (metals, molecules, etc.) in the turbulent interstellar medium (ISM) is critical for abundance patterns of stars and clusters, galaxy and star formation, and cooling from the circumgalactic medium. However, the fundamental scaling laws remain poorly understood (and usually unresolved in numerical simulations) in the highly supersonic, magnetized, shearing regime relevant for the ISM.We therefore study the full scaling laws governing passive-scalar transport in idealized simulations of supersonic MHD turbulence, including shear. Using simple phenomenological arguments for the variation of diffusivity with scale based on Richardson diffusion, we propose a simple fractional diffusion equation to describe the turbulent advection of an initial passive scalar distribution. These predictions agree well with the measurements from simulations, and vary with turbulent Mach number in the expected manner, remaining valid even in the presence of a large-scale shear flow (e.g. rotation in a galactic disk). The evolution of the scalar distribution is not the same as obtained using simple, constant “effective diffusivity” as in Smagorinsky models, because the scale-dependence of turbulent transport means an initially Gaussian distribution quickly develops highly non-Gaussian tails. We also emphasize that these are mean scalings that only apply to ensemble behaviors (assuming many different, random scalar injection sites): individual Lagrangian “patches” remain coherent (poorly-mixed) and simply advect for a large number of turbulent flow-crossing times.

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M. Colbrook, X. Ma, P. Hopkins, et. al.
Mon, 24 Oct 16

Comments: submitted to MNRAS, 8 pages, 4 figures, comments welcome

Symplectic fourth-order maps for the collisional N-body problem [CL]


We study analytically and experimentally certain symplectic and time-reversible N-body integrators which employ a Kepler solver for each pair-wise interaction, including the method of Hernandez & Bertschinger (2015). Owing to the Kepler solver, these methods treat close two-body interactions correctly, while close three-body encounters contribute to the truncation error at second order and above. The second-order errors can be corrected to obtain a fourth-order scheme with little computational overhead. We generalise this map to an integrator which employs a Kepler solver only for selected interactions and yet retains fourth-order accuracy without backward steps. In this case, however, two-body encounters not treated via a Kepler solver contribute to the truncation error.

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W. Dehnen and D. Hernandez
Fri, 30 Sep 16

Comments: 17 pages, re-submitted to MNRAS

Hybrid Entropy Stable HLL-Type Riemann Solvers for Hyperbolic Conservation Laws [CL]


It is known that HLL-type schemes are more dissipative than schemes based on characteristic decompositions. However, HLL-type methods offer greater flexibility to large systems of hyperbolic conservation laws because the eigenstructure of the flux Jacobian is not needed. We demonstrate in the present work that several HLL-type Riemann solvers are provably entropy stable. Further, we provide convex combinations of standard dissipation terms to create hybrid HLL-type methods that have less dissipation while retaining entropy stability. The decrease in dissipation is demonstrated for the ideal MHD equations with a numerical example.

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B. Schmidtmann and A. Winters
Fri, 22 Jul 16

Comments: 5 pages

A Hybrid Riemann Solver for Large Hyperbolic Systems of Conservation Laws [CL]


We are interested in the numerical solution of large systems of hyperbolic conservation laws or systems in which the characteristic decomposition is expensive to compute. Solving such equations using finite volumes or Discontinuous Galerkin requires a numerical flux function which solves local Riemann problems at cell interfaces. There are various methods to express the numerical flux function. On the one end, there is the robust but very diffusive Lax-Friedrichs solver; on the other end the upwind Godunov solver which respects all resulting waves. The drawback of the latter method is the costly computation of the eigensystem.
This work presents a family of simple first order Riemann solvers, named HLLX$\omega$, which avoid solving the eigensystem. The new method reproduces all waves of the system with less dissipation than other solvers with similar input and effort, such as HLL and FORCE. The family of Riemann solvers can be seen as an extension or generalization of the methods introduced by Degond et al. \cite{DegondPeyrardRussoVilledieu1999}. We only require the same number of input values as HLL, namely the globally fastest wave speeds in both directions, or an estimate of the speeds. Thus, the new family of Riemann solvers is particularly efficient for large systems of conservation laws when the spectral decomposition is expensive to compute or no explicit expression for the eigensystem is available.

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B. Schmidtmann and M. Torrilhon
Thu, 21 Jul 16

Comments: arXiv admin note: text overlap with arXiv:1606.08040