Une m�thode multipas implicite-explicite pour l'approximation des �quations d'�volution paraboliques

Crouzeix, Michel

doi:10.1007/bf01396412

Cited by 136 publications

(141 citation statements)

References 2 publications

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“…For derivations of these schemes, see [1,6,30]. The absolute stability regions of the explicit halves of some time integration schemes are shown in Figure 3.1 for reference.…”

Section: Time Stepping Using Green's Functionsmentioning

confidence: 99%

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Viswanath

Tobasco

2013

Journal of Computational Physics

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Numerical solvers of the incompressible Navier-Stokes equations have reproduced turbulence phenomena such as the law of the wall, the dependence of turbulence intensities on the Reynolds number, and experimentally observed properties of turbulence energy production. In this article, we begin a sequence of investigations whose eventual aim is to derive and implement numerical solvers that can reach higher Reynolds numbers than is currently possible. Every time step of a Navier-Stokes solver in effect solves a linear boundary value problem. The use of Green's functions leads to numerical solvers which are highly accurate in resolving the boundary layer, which is a source of delicate but exceedingly important physical effects at high Reynolds numbers. The use of Green's functions brings with it a need for careful quadrature rules and a reconsideration of time steppers. We derive and implement Green's function based solvers for the channel flow and plane Couette flow geometries. The solvers are validated by reproducing turbulence phenomena in good agreement with earlier simulations and experiment.

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“…For derivations of these schemes, see [1,6,30]. The absolute stability regions of the explicit halves of some time integration schemes are shown in Figure 3.1 for reference.…”

Section: Time Stepping Using Green's Functionsmentioning

confidence: 99%

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Viswanath

Tobasco

2013

Journal of Computational Physics

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“…In view of the fact that H(x) 2 , H(x) −1 2 are uniformly bounded, see relations (3.27) and (3.28) in [3], it suffices to estimate Y n . We adjust in this section the definition of ||| · ||| to the scheme under consideration by setting…”

Section: Multistep Schemesmentioning

confidence: 99%

“…Following an idea of Crouzeix, [3], for the time discretization of parabolic equations with time dependent coefficients, we combine implicit and explicit multistep schemes to discretize (1.2) in time: Implicit schemes are used for discretizing the left-hand side of the o.d.e. in (1.2), and explicit schemes for the nonlinear righthand side.…”

Section: Introductionmentioning

confidence: 99%

Implicit-explicit multistep finite element methods for nonlinear parabolic problems

1998

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Abstract. We approximate the solution of initial boundary value problems for nonlinear parabolic equations. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient, since their implementation requires at each time step the solution of a linear system with the same matrix for all time levels. We derive optimal order error estimates. The abstract results are applied to the Kuramoto-Sivashinsky and the CahnHilliard equations in one dimension, as well as to a class of reaction diffusion equations in R ν , ν = 2, 3.

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“…A splitting approach offers a reduced degree of computational complexity and thus straightforward implementations with respect to IMEX methods, which require additional stability and order conditions that combine all inner implicit-explicit schemes [21,22,23,24]. However, appropriate criteria must be introduced to efficiently decouple the physical phenomena via splitting and to control the so-called splitting errors [25].…”

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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Une m�thode multipas implicite-explicite pour l'approximation des �quations d'�volution paraboliques

Cited by 136 publications

References 2 publications

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Implicit-explicit multistep finite element methods for nonlinear parabolic problems

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

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