Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations

Caramana, E.J.; Shashkov, Mikhail; Whalen, P.P.

doi:10.1006/jcph.1998.5989

Cited by 250 publications

(267 citation statements)

References 18 publications

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“…Here, we choose to compute it according to the Kuropatenko formula (see, e.g., Caramana et al 1998) and add it as an extra pressure term in both the momentum and energy equations. To model the behavior of the two different species we choose to add the artificial viscosity term to the proton energy equation and to leave unchanged the electron pressure equation.…”

Section: Modeling Approachmentioning

confidence: 99%

Plasma Physical Parameters Along Cme-Driven Shocks. Ii. Observation–simulation Comparison

Bacchini

Susino

Бемпорад

et al. 2015

ApJ

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In this work, we compare the spatial distribution of the plasma parameters along the 1999 June 11 coronal mass ejection (CME)-driven shock front with the results obtained from a CME-like event simulated with the FLIPMHD3D code, based on the FLIP-MHD particle-in-cell method. The observational data are retrieved from the combination of white-light coronagraphic data (for the upstream values) and the application of the RankineHugoniot equations (for the downstream values). The comparison shows a higher compression ratio X and Alfvénic Mach number M A at the shock nose, and a stronger magnetic field deflection d toward the flanks, in agreement with observations. Then, we compare the spatial distribution of M A with the profiles obtained from the solutions of the shock adiabatic equation relating M A , X, and Bn q (the angle between the upstream magnetic field and the shock front normal) for the special cases of parallel and perpendicular shock, and with a semi-empirical expression for a generically oblique shock. The semi-empirical curve approximates the actual values of M A very well, if the effects of a non-negligible shock thickness sh d and plasma-to magnetic pressure ratio u b are taken into account throughout the computation. Moreover, the simulated shock turns out to be supercritical at the nose and sub-critical at the flanks. Finally, we develop a new one-dimensional Lagrangian ideal MHD method based on the GrAALE code, to simulate the ion-electron temperature decoupling due to the shock transit. Two models are used, a simple solar wind model and a variable-γ model. Both produce results in agreement with observations, the second one being capable of introducing the physics responsible for the additional electron heating due to secondary effects (collisions, Alfvén waves, etc.).

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Section: Modeling Approachmentioning

confidence: 99%

Plasma Physical Parameters Along Cme-Driven Shocks. Ii. Observation–simulation Comparison

Bacchini

Susino

Бемпорад

et al. 2015

ApJ

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“…In the present paper, we further incorporate the radiation force. We also adopt an artificial viscosity formulation of Caramana et al (1998), designed to distinguish between shock-wave and uniform compression using an advection limiter.…”

Section: Numerical Schemementioning

confidence: 99%

The Structure and Evolution of Early Cosmological HiiRegions

Kitayama

Yoshida

Susa

et al. 2004

ApJ

276

322

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We study the formation and evolution of H ii regions around the first stars formed at redshifts z ¼ 10 30. We use a one-dimensional Lagrangian hydrodynamics code that self-consistently incorporates radiative transfer and nonequilibrium primordial gas chemistry. The star-forming region is defined as a spherical dense molecular gas cloud with a Population III star embedded at the center. We explore a large parameter space by considering, as plausible early star-forming sites, dark matter halos of mass M halo ¼ 10 5 10 8 M , gas density profiles with a power-law index w ¼ 1:5 2:25, and metal-free stars of mass M star ¼ 25 500 M . The formation of the H ii region is characterized by initial slow expansion of a weak D-type ionization front near the center, followed by rapid propagation of an R-type front throughout the outer gas envelope. We find that the transition between the two front types is indeed a critical condition for the complete ionization of halos of cosmological interest. In small-mass (P10 6 M ) halos, the transition takes place within a few 10 5 yr, yielding high escape fractions (>80%) of both ionizing and photodissociating photons. The gas is effectively evacuated by a supersonic shock, with the mean density within the halo decreasing to P1 cm À3 in a few million years. In larger mass (k10 7 M ) halos, the ionization front remains to be of D-type over the lifetime of the massive star, the H ii region is confined well inside the virial radius, and the escape fractions are essentially zero. We derive an analytic formula that reproduces well the results of our simulations for the critical halo mass below which the gas is completely ionized. We discuss immediate implications of the present results for the star formation history and early reionization of the universe.

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“…This question has been investigated by other authors in a number of ways, but here we seek the answer through a detailed modified equation analysis. If one takes the stance that the modified equation is the PDE which the numerics solve more accurately than the original PDE (presumably a small perturbation to the original problem), then the quest becomes one of understanding the behavior of (4) and (5). Both of these modified equations can be viewed as approximations to the advection-diffusion equation…”

Section: Analysis Of the Problemmentioning

confidence: 99%

“…Certainly for Eulerian schemes [2,3], all waves are captured and so the connection is clear. Lagrangian schemes [4][5][6] are often used to circumvent the issues associated with captured linear jumps by performing computations in the frame of the fluid, but most interesting simulations require, at the very least, mesh remap which results in the so-called arbitrary-Lagrangian-Eulerian (ALE) schemes [7]. The use of such remap has the potential to cause ALE schemes suffer the same fate as purely Eulerian schemes although the details may depend on the frequency at which remapping is performed.…”

Section: Introductionmentioning

confidence: 99%

On sub-linear convergence for linearly degenerate waves in capturing schemes

Banks

Aslam

Rider

2008

Journal of Computational Physics

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A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2 , yielding a convergence rate of 1/2. Self-similar heuristics for higher order discretizations produce a growth rate for the layer thickness of ∆t 1/(p+1) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.

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Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations

Cited by 250 publications

References 18 publications

Plasma Physical Parameters Along Cme-Driven Shocks. Ii. Observation–simulation Comparison

Plasma Physical Parameters Along Cme-Driven Shocks. Ii. Observation–simulation Comparison

The Structure and Evolution of Early Cosmological HiiRegions

On sub-linear convergence for linearly degenerate waves in capturing schemes

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