Local Rectangular Refinement with Application to Nonreacting and Reacting Fluid Flow Problems

Bennett, Beth Anne V.; Smooke, Mitchell D.

doi:10.1006/jcph.1999.6214

Cited by 48 publications

(73 citation statements)

References 44 publications

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“…The lid, moving with a steady and spatially uniform velocity, sets up a principal vortex and subsidiary corner vortices by viscous forces and the differentially heated lateral walls of the cavity set up a buoyant vortex flow, which opposes the principal lid-driven vortex. Our parameterization and discretization are inspired by [2]; however, in this example we do not exploit local adaptive refinement. The PETSc implementation contains uniform-refinement mesh-sequencing capabilities, applied both in a nonlinear continuation sense in the outer iteration, as described in the introduction, and as an inner iteration as part of a multigrid solver.…”

Section: Incompressible Flowmentioning

confidence: 99%

“…In the interests of space, we leave the derivation of this velocity-vorticity form of the Navier-Stokes and energy governing equations (3.1)-(3.4) to the literature (e.g., [2] and citations therein). However, to motivate our main point, we note that the two equations with velocity components under the Laplacian operators come from differentiating the continuity equation and substituting from the definition of vorticity.…”

Section: Incompressible Flowmentioning

confidence: 99%

See 1 more Smart Citation

Pseudotransient Continuation and Differential-Algebraic Equations

Coffey¹,

Kelley²,

Keyes³

2003

SIAM J. Sci. Comput.

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Abstract. Pseudo-transient continuation is a practical technique for globalizing the computation of steady-state solutions of nonlinear differential equations. The technique employs adaptive time-stepping to integrate an initial value problem derived from an underlying ODE or PDE boundary value problem until sufficient accuracy in the desired steady-state root is achieved to switch over to Newton's method and gain a rapid asymptotic convergence. The existing theory for pseudo-transient continuation includes a global convergence result for differential equations written in semidiscretized method-of-lines form. However, many problems are better formulated or can only sensibly be formulated as differentialalgebraic equations (DAEs). These include systems in which some of the equations represent algebraic constraints, perhaps arising from the spatial discretization of a PDE constraint.Multirate systems, in particular, are often formulated as differential-algebraic systems to suppress fast time scales (acoustics, gravity waves, Alfven waves, near equilibrium chemical oscillations, etc.) that are irrelevant on the dynamical time scales of interest. In this paper we present a global convergence result for pseudo-transient continuation applied to DAEs of index 1, and we illustrate it with numerical experiments on model incompressible flow and reacting flow problems, in which a constraint is employed to step over acoustic waves.

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Section: Incompressible Flowmentioning

confidence: 99%

Section: Incompressible Flowmentioning

confidence: 99%

Pseudotransient Continuation and Differential-Algebraic Equations

Coffey¹,

Kelley²,

Keyes³

2003

SIAM J. Sci. Comput.

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“…In order to verify that the accuracy of the computations is settled by the accuracy tolerances, one may define for problem (13), discretized on a uniform mesh of 1024 2 : • A quasi-exact reference solution c J qe , obtained with the Strang scheme (7) with a small and constant splitting time step of ∆t = 10 −7 ;…”

Section: Characterization Of the Numerical Performance Of The Methodsmentioning

confidence: 99%

“…This approach can then be coupled with other modeling considerations, like those presented previously, and with a suitable exploitation of modern computing capabilities. For instance, implicit time integration methods are investigated for complex applications [13,14,15,16] 2 , and efforts are also being made on computationally less demanding hybrid methods which combine implicit and explicit schemes such as IMEX [17,18] or operator splitting [19,20] techniques.…”

Section: Introductionmentioning

confidence: 99%

Time–space adaptive numerical methods for the simulation of combustion fronts

Duarte¹,

Descombes²,

Tenaud³

et al. 2013

Combustion and Flame

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To cite this version:Max Duarte, Stéphane Descombes, Christian Tenaud, Sébastien Candel, Marc Massot. Time-space adaptive numerical methods for the simulation of combustion fronts. Combustion and Flame, Elsevier, 2013, 160 (6) AbstractThis paper presents a new computational strategy for the simulation of combustion fronts based on adaptive time operator splitting and spatial multiresolution. High-order and dedicated onestep solvers compose the splitting scheme for the reaction, diffusion, and convection subproblems, to independently cope with their inherent numerical difficulties and to properly solve the corresponding temporal scales. Adaptive and thus highly compressed spatial representations for localized fronts originating from multiresolution analysis result in important reductions of memory usage, and hence numerical simulations with sufficiently fine spatial resolution can be performed with standard computational resources. The computational efficiency is further enhanced by splitting time steps established beyond standard stability constraints associated to mesh size or stiff source time scales. The splitting time steps are chosen according to a dynamic splitting technique relying on solid mathematical foundations, which ensures error control of the time integration and successfully discriminates time-varying multi-scale physics. For a given semidiscretized problem, the solution scheme provides dynamic accuracy estimates that reflect the quality of numerical results in terms of numerical errors of integration and compressed spatial representations, for general multi-dimensional problems modeled by stiff PDEs. The strategy is efficiently applied to simulate the propagation of laminar premixed flames interacting with vortex structures, as well as various configurations of self-ignition processes of diffusion flames in similar vortical hydrodynamics fields. A detailed study of the error control is provided and show the potential of the approach. It yields large gains in CPU time, while consistently describing a broad spectrum of space and time scales as well as different physical scenarios.

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“…When solving such problems numerically, this solution behaviour requires a much finer grid in these high activity regions than in regions where the solution is fairly smooth. This is the case, for instance, for the equations that describe laminar flames [1]: most of the activity is concentrated in the 'flame front', a narrow area where the temperature rises steeply and chemical reactions take place. Finite element methods can quite easily cope with this, but for finite difference or finite volume methods the unstructured grids, that are a consequence of this non-uniform behaviour, are much more complicated to handle.…”

Section: Introductionmentioning

confidence: 99%

Local defect correction with slanting grids

Graziadei

Mattheij

Boonkkamp

2003

Numerical Methods Partial

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The local defect correction (LDC) method is used to solve a convection‐diffusion‐reaction problem that contains a high‐activity region in a relatively small part of the domain. The improvement of the solution on a coarse grid is obtained by introducing a correction term computed from a local fine‐grid solution. This article studies problems where the high‐activity region is covered with a rectangular fine grid not aligned with the axes of the global domain. This study shows that the resulting method is less expensive than both a uniform refinement and tensor product grid method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 1–17, 2004.

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Local Rectangular Refinement with Application to Nonreacting and Reacting Fluid Flow Problems

Cited by 48 publications

References 44 publications

Pseudotransient Continuation and Differential-Algebraic Equations

Pseudotransient Continuation and Differential-Algebraic Equations

Time–space adaptive numerical methods for the simulation of combustion fronts

Local defect correction with slanting grids

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