Convergence  of a  step-doubling Galerkin method  for parabolic problems

Ayati, Bruce P.; Dupont, Todd

doi:10.1090/s0025-5718-04-01696-5

Cited by 15 publications

(13 citation statements)

References 12 publications

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“…Equation (14) (or equations (9)- (10) with accompanying conditions) along with equations (6) and (7) are solved numerically. We use a moving-grid Galerkin method in age with discontinuous piecewise linear functions post-processed to cubic splines for the approximation space in age (Ayati & Dupont, 2002) along with a step-doubling method in time (Ayati & Dupont, 2005). This combination was illustrated in Section 5, Ayati & Dupont (2002), using the same code as used here.…”

Section: Antimicrobial Parameters (Except D)mentioning

confidence: 99%

Senescence can explain microbial persistence

et al. 2007

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It has been known for many years that small fractions of persister cells resist killing in many bacterial colony-antimicrobial confrontations. These persisters are not believed to be mutants. Rather it has been hypothesized that they are phenotypic variants. Current models allow cells to switch in and out of the persister phenotype. Here, a different explanation is suggested for persistence, namely senescence. Using a mathematical model including age structure, it is shown that senescence provides a natural explanation for persistence-related phenomena, including the observations that the persister fraction depends on growth phase in batch culture and dilution rate in continuous culture.

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Section: Antimicrobial Parameters (Except D)mentioning

confidence: 99%

Senescence can explain microbial persistence

et al. 2007

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“…In order to achieve second-order accuracy in time, we use a scheme analyzed in detail in [14]. We compute two different approximate solutions : the first one is w (1) n+1 for one step δt, the second is w (2) n+1 for two half steps of size δt/2.…”

Section: Richardson Extrapolationmentioning

confidence: 99%

“…The method is therefore of second order, in the sense that the accumulated error scales like O(δt 2 ). When used in conjunction with a backward Euler step, the stability properties of this method are also very favorable [14].…”

Section: Richardson Extrapolationmentioning

confidence: 99%

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The Explicit–Implicit–Null method: Removing the numerical instability of PDEs

Duchemin

Eggers

2014

Journal of Computational Physics

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A general method to remove the numerical instability of partial differential equations is presented. Two equal terms are added to and subtracted from the right-hand-side of the PDE : the first is a damping term and is treated implicitly, the second is treated explicitly. A criterion for absolute stability is found and the scheme is shown to be convergent. The method is applied with success to the mean curvature flow equation, the Kuramoto-Sivashinsky equation, and to the Rayleigh-Taylor instability in a Hele-Shaw cell, including the effect of surface tension.

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“…Many papers deal with the Richardson extrapolation, which is also referred to as the doubling method or h 2 -/h 4 -extrapolation. Hence, considerations for using the doubling method can be for instance, the mathematical convergence aspects (Ayati andDupont 2004,Aïd 1999), the increase of accuracy order (Abbasian and Carey 1997) or the applicability to both time and spatial grids (Richards 1997).…”

Section: Introductionmentioning

confidence: 99%

Implementation of Richardson extrapolation in an efficient adaptive time stepping method: applications to reactive transport and unsaturated flow in porous media

2007

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Environmental studies are commonly carried out through numerical simulations, which have to be accurate, reliable and efficient. When transient problems are considered, the validity of the solutions requires the calculation and management of the temporal discretization errors. This article describes an adaptive time stepping strategy based on the estimation of the local truncation error via the Richardson extrapolation technique. The time-marching scheme is mathematically based on this a posteriori error estimation that has to be gauged. General optimizations are also suggested making the control of both the temporal error and the evolution of the time step size very efficient. Furthermore, the algorithm connecting these methods is all the more interesting as it could be implemented in many computational codes using different numerical schemes. In the hydrogeochemical domain, this algorithm represents an interesting alternative to a fixed time step as shown by the various numerical tests involving reactive transport and unsaturated flow.

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Convergence of a step-doubling Galerkin method for parabolic problems

Cited by 15 publications

References 12 publications

Senescence can explain microbial persistence

Senescence can explain microbial persistence

The Explicit–Implicit–Null method: Removing the numerical instability of PDEs

Implementation of Richardson extrapolation in an efficient adaptive time stepping method: applications to reactive transport and unsaturated flow in porous media

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