A Rapid Approximation Procedure for Nonlinear Solutions: Application to Aerodynamic Flows

AbstractWe prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

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Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Acta Numerica ◽

10.1017/s0962492920000057 ◽

2020 ◽

Vol 29 ◽

pp. 701-762

Author(s):

Chi-Wang Shu

Keyword(s):

Main Idea ◽

High Order Accuracy ◽

Finite Difference Schemes ◽

Approximation Procedure ◽

Discontinuous Solutions ◽

Weno Schemes ◽

Convection Diffusion ◽

Different Types ◽

Basic Ideas ◽

Weighted Eno

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

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Computing the Sound of the Sea in a Seashell

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09509-9 ◽

2021 ◽

Author(s):

Jonathan Ben-Artzi ◽

Marco Marletta ◽

Frank Rösler

Keyword(s):

Characteristic Function ◽

Helmholtz Resonator ◽

Positive Answer ◽

Approximation Procedure ◽

Universal Algorithm

AbstractThe question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $${{\mathcal {C}}}^2$$ C 2 . The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

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Computational study of a novel microwave excited plasma sensor for aerodynamic flows

Journal of Applied Physics ◽

10.1063/5.0037743 ◽

2021 ◽

Vol 129 (8) ◽

pp. 084503

Author(s):

Anand Karpatne ◽

Ashish Sharma ◽

Jayant Sirohi ◽

Laxminarayan L. Raja

Keyword(s):

Computational Study ◽

Aerodynamic Flows

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Interactions between steady and oscillatory convection in mushy layers

Journal of Fluid Mechanics ◽

10.1017/s0022112009992709 ◽

2010 ◽

Vol 645 ◽

pp. 411-434 ◽

Cited By ~ 13

Author(s):

PETER GUBA ◽

M. GRAE WORSTER

Keyword(s):

Standing Waves ◽

Mushy Layer ◽

Two Dimensional ◽

Oscillatory Convection ◽

Mushy Layers ◽

Theoretical Predictions ◽

Convection Patterns ◽

The Stability ◽

Nonlinear Solutions ◽

Steady Convection

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

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Photoconductivity of Low-Bandgap Polymer and Polymer: Fullerene Bulk Heterojunction Studied by Constant Photocurrent Method

Organic Photonics and Photovoltaics ◽

10.1515/oph-2015-0011 ◽

2015 ◽

Vol 3 (1) ◽

Cited By ~ 2

Author(s):

V. V. Malov ◽

A. R. Tameev ◽

S. V. Novikov ◽

M. V. Khenkin ◽

A. G. Kazanskii ◽

...

Keyword(s):

Molecular Orbital ◽

Bulk Heterojunction ◽

Inorganic Materials ◽

Fullerene Derivative ◽

Approximation Procedure ◽

Photoelectric Properties ◽

Highest Occupied Molecular Orbital ◽

Donor Acceptor ◽

Lowest Unoccupied Molecular Orbital ◽

Doped Polymers

AbstractOptical and photoelectric properties of modern photosensitive polymers are of great interest due to their prospects for photovoltaic applications. In particular, an investigation of absorption and photoconductivity edge of these materials could provide valuable information. For these purpose we applied the constant photocurrent method which has proved its efficiency for inorganic materials. PCDTBT and PTB7 polymers were used as objects for the study as well as their blends with a fullerene derivative PC71BM. The measurements by constant photocurrent method (CPM) show that formation of bulk heterojunction (BHJ) in the blends increases photoconductivity and results in a redshift of the photocurrent edge in the doped polymers compared with that in the neat polymers. Obtained from CPM data, spectral dependences of absorption coefficient were approximated using Gaussian distribution of density-of-states within HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) bands. The approximation procedure allowed us to evaluate rather optical than electrical bandgaps for the studied materials. Moreover, spectra of polymer:PC71BM blends were fitted well by the sum of two Gaussian peaks which reveal both the transitions within the polymer and the transitions involving charge transfer states at the donor-acceptor interface in the BHJ.

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