A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

Martín, Daniel; Colella, Phillip; Graves, Daniel

doi:10.1016/j.jcp.2007.09.032

Cited by 72 publications

(77 citation statements)

References 29 publications

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“…et al 2012) along with coupled self-gravity (Martin et al 2008) and radiation transfer (Krumholz et al 2007) in the two-temperature, mixedframe, grey, flux-limited diffusion approximation. orion2 utilizes a conservative second order Godunov scheme to solve the equations of compressible gas dynamics (Truelove et al 1998;Klein 1999).…”

Section: Methodsmentioning

confidence: 99%

Formation of stellar clusters in magnetized, filamentary infrared dark clouds

Klein

McKee

2017

Monthly Notices of the Royal Astronomical Society

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Star formation in a filamentary infrared dark cloud (IRDC) is simulated over a dynamic range of 4.2 pc to 28 au for a period of 3.5 × 10 5 yr, including magnetic fields and both radiative and outflow feedback from the protostars. At the end of the simulation, the star formation efficiency is 4.3 per cent and the star formation rate per free fall time is ff 0.04, within the range of observed values (Krumholz et al. 2012a). The total stellar mass increases as ∼ t 2 , whereas the number of protostars increases as ∼ t 1.5 . We find that the density profile around most of the simulated protostars is ∼ ρ ∝ r −1.5 , as predicted by . At the end of the simulation, the protostellar mass function approaches the Chabrier (2005) stellar initial mass function. We infer that the time to form a star of median mass 0.2 M is about 1.4 × 10 5 yr from the median mass accretion rate. We find good agreement among the protostellar luminosities observed in the large sample of Dunham et al. (2013), our simulation, and a theoretical estimate, and conclude that the classical protostellar luminosity problem (Kenyon et al. 1990) is resolved. The multiplicity of the stellar systems in the simulation agrees to within a factor 2 of observations of Class I young stellar objects; most of the simulated multiple systems are unbound. Bipolar protostellar outflows are launched using a sub-grid model, and extend up to 1 pc from their host star. The mass-velocity relation of the simulated outflows is consistent with both observation and theory.

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Section: Methodsmentioning

confidence: 99%

Formation of stellar clusters in magnetized, filamentary infrared dark clouds

Klein

McKee

2017

Monthly Notices of the Royal Astronomical Society

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“…For the collocated discretization, we solve the incompressible Navier-Stokes equations by either an exact [12][13][14][15] or an approximate [16][17][18][19][20][21] projection method. Exact projection methods ensure that the discrete divergence of the Eulerian velocity field is zero to machine accuracy (when direct solvers are used) or to within the tolerance of the linear solver (when iterative solvers are used).…”

Section: Introductionmentioning

confidence: 99%

On the Volume Conservation of the Immersed Boundary Method

Griffith

2012

Commun. comput. phys.

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Abstract.The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests considered herein, the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes. We also compare the performance of cell-centered schemes that use either exact or approximate projection methods. We find that the volumeconservation properties of approximate projection IB methods depend strongly on the formulation of the projection method. When used with the IB method, we find that pressure-free approximate projection methods can yield extremely poor volume conservation, whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.

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“…Typically the cells to be changed are "tagged" according to the evaluation of certain criteria; these criteria, specified by the user, are usually based on physical quantities or estimated errors. This dynamic grid generation is done in the standard fashion described in [2,13], by averaging down from finer grids to coarser grids as the former disappear, and interpolating conservatively from coarser grids to newly refined regions. The principal difference in the present AMR algorithm is the higher-order, conservative coarse-fine interpolation and its supporting multigrid-based solver, as described above and in section 2.3.…”

Section: Time Integrationmentioning

confidence: 99%

“…The uniform grid discretization above can also be extended to structured adaptive mesh refinement (AMR), i.e., a locally refined, nested hierarchy of rectangular grids. Our notation is based on previous O(h 2 ) structured AMR work (see [13]), but we will reiterate parts of the notation for the purpose of explaining the present algorithm.…”

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confidence: 99%

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A Fourth-Order Accurate Finite-Volume Method with Structured Adaptive Mesh Refinement for Solving the Advection-Diffusion Equation

Zhang¹,

Johansen²,

Colella³

2012

SIAM J. Sci. Comput.

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Abstract. We present a fourth-order accurate algorithm for solving Poisson's equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditions; the resulting linear system is solved with a multigrid algorithm. For time integration, we couple the elliptic solver to a fourth-order accurate Runge-Kutta method, introduced by Kennedy and Carpenter [Appl. Numer. Math., 44 (2003), pp. 139-181], which enables us to treat the nonstiff advection term explicitly and the stiff diffusion term implicitly. We demonstrate the spatial and temporal accuracy by comparing results with analytical solutions. Because of the general formulation of the approach, the algorithm is easily extensible to more complex physical systems. 1. Introduction. The advection-diffusion equation governs numerous physical processes. Morton [17] lists ten sample applications ranging from semiconductor simulation to financial modelling. He also observes that "Accurate modelling of the interaction between convective and diffusive processes is the most ubiquitous and challenging task in the numerical approximation of partial differential equations." This observation is partly due to the fact that algorithms and analysis tend to be very different in the two limiting cases of elliptic and hyperbolic equations. Also, even for very simple initial and boundary conditions, the true solution may contain multiple length-scales that vary drastically across the spatial domain; see (3.1) in [18] for such an example.A finite-volume (FV) formulation is often preferred for applications where conservation is a primary concern. In its simplest form, the FV formulation is derived by applying the divergence theorem over the cells of a regular computational grid:

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A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions

Cited by 72 publications

References 29 publications

Formation of stellar clusters in magnetized, filamentary infrared dark clouds

Formation of stellar clusters in magnetized, filamentary infrared dark clouds

On the Volume Conservation of the Immersed Boundary Method

A Fourth-Order Accurate Finite-Volume Method with Structured Adaptive Mesh Refinement for Solving the Advection-Diffusion Equation

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