An accurate finite-difference method for ablation-type Stefan problems

Mitchell, Sarah; Vynnycky, Michael

doi:10.1016/j.cam.2012.05.011

Cited by 36 publications

(11 citation statements)

References 28 publications

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“…10. See also the related experimental paper [71] VI. DISCUSSION Our proposed model indicates that the NIC design is located near a performance cliff, a conclusion consistent with NIF/NIC experimental data.…”

Section: Combined Effects: Ablation and Dt To Ch Rt Instabilitiesmentioning

confidence: 99%

Mixing with applications to inertial-confinement-fusion implosions

et al. 2017

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Approximate 1D as well as 2D and 3D simulations are playing an important supporting role in the design and analysis of future experiments at NIF. This paper is mainly concerned with 1D simulations, used extensively in design and optimization. We couple a 1D buoyancy-drag mix model for the mixing zone edges with a 1D Inertial Confinement Fusion (ICF) simulation code. This analysis predicts that National Ignition Campaign designs are located close to a performance cliff, so that modeling errors, design features (fill tube and tent) and additional, unmodeled instabilities could lead to significant levels of mix. The performance cliff we identify is associated with multimode plastic ablator (CH) mix into the hot spot Deuterium and Tritium (DT). The buoyancy-drag mix model is mode number independent, and selects implicitly a range of maximum growth modes.Our main conclusion is that single effect instabilities are predicted not to lead to hot spot mix, while combined mode mixing effects are predicted to affect hot spot thermodynamics and possibly hot spot mix. Combined with the stagnation Rayleigh-Taylor instability, we find the potential for mix effects in combination with the ice to gas DT boundary, numerical effects of Eulerian species CH concentration diffusion and ablation driven instabilities.With the help of a convenient package of plasma transport parameters developed here, we give an approximate determination of these quantities in the regime relevant to the NIC experiments, while ruling out a variety of mix possibilities. Plasma transport parameters affect the 1D buoyancy-drag mix model primarily through its phenomenological drag coefficient as well as the 1D hydro model the buoyancy-drag equation is coupled to.

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Section: Combined Effects: Ablation and Dt To Ch Rt Instabilitiesmentioning

confidence: 99%

Mixing with applications to inertial-confinement-fusion implosions

et al. 2017

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“…The first concerns the formal accuracy of the numerical scheme used to solve the governing equations; this appears never to have been considered. It turns out to be of relevance in view of our recent work using the formally second-order accurate Keller Box finite-difference method for moving boundary problems, where we showed that the actual accuracy of a numerical scheme decreases if inconsistencies between initial and boundary conditions are not treated correctly [23][24][25]; indeed, this is exactly what happens in the oxygen diffusion problem. The second aspect that we consider is the actual value of the time, t e , at which the oxygen is depleted; we shall call this quantity the extinction time, in line with other physical situations where a domain that is initially of finite extent vanishes after a finite time [26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 88%

“…Moreover, as explained in [23][24][25], consistency is important because it has an effect on the accuracy of the numerical scheme that is used; from this point of view, it is advantageous to apply a scheme to a formulation where both the dependent variable and its spatial derivative are consistent at both (0,0) and (1,0). A natural first step towards achieving this is to define a new dependent variable, u, given by…”

Section: Transformations Leading To a Consistent Formmentioning

confidence: 99%

“…In Section 3, we extract important analytical details from the problem, both in the small-time limit and as the oxygen is depleted. Section 4 explains how the resulting equations are implemented numerically using the analysis presented in Section 3; as in [23][24][25]32], we use the Keller box scheme in tandem with the boundary immobilization method. The results are then presented and discussed in Section 5, and conclusions are drawn in Section 6.…”

Section: Introductionmentioning

confidence: 99%

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The oxygen diffusion problem: Analysis and numerical solution

Mitchell

Vynnycky

2015

Applied Mathematical Modelling

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Please cite this article as: S.L. Mitchell, M. Vynnycky, The oxygen diffusion problem: analysis and numerical solution, Appl. Math. Modelling (2014), doi: http://dx. AbstractA recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is applied to the classical moving boundary problem that arises from the diffusion of oxygen in absorbing tissue; in tandem with the Keller box finite-difference scheme, the so-called boundary immobilization method is used. New insights are obtained into three aspects of the problem: the numerical accuracy of the scheme used; the calculation of oxygen depletion time; and the behaviour of the moving boundary as the oxygen is depleted.

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“…The above equations were then solved numerically using the Keller box finite-difference method, whose implementation for this type of initially degenerate moving-boundary problem has been documented elsewhere (see [23,24]); all definite integrals that arise in the formulation were computed using the trapezoidal rule. The results of the numerical simulations are shown and discussed in Section 4.…”

Section: Model Equations For Numerical Simulationmentioning

confidence: 99%