Exact solution of the proximity effect equation by a splitting method

1988 ◽  
Vol 6 (1) ◽  
pp. 432 ◽  
Author(s):  
P. Dean Gerber
Keyword(s):  
Exact Solution ◽  
Proximity Effect ◽  
Splitting Method

scholarly journals Assessment of Hydrometeor Collection Rates from Exact and Approximate Equations. Part II: Numerical Bounding

2007 ◽  
Vol 46 (1) ◽  
pp. 82-96
Author(s):  
Brian J. Gaudet ◽  
Jerome M. Schmidt
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Splitting Method ◽  
Time Discretization ◽  
Large Mass ◽  
Time Step ◽  
Transfer Rates ◽  
Mass Moment ◽  
Time Splitting ◽  
Approximate Equations

Abstract Past microphysical investigations, including Part I of this study, have noted that the collection equation, when applied to the interaction between different hydrometeor species, can predict large mass transfer rates, even when an exact solution is used. The fractional depletion in a time step can even exceed unity for the collected species with plausible microphysical conditions and time steps, requiring “normalization” by a microphysical scheme. Although some of this problem can be alleviated through the use of more moment predictions and hydrometeor categories, the question as to why such “overdepletion” can be predicted in the first place remains insufficiently addressed. It is shown through both physical and conceptual arguments that the explicit time discretization of the bulk collection equation for any moment is not consistent with a quasi-stochastic view of collection. The result, under certain reasonable conditions, is a systematic overprediction of collection, which can become a serious error in the prediction of higher-order moments in a bulk scheme. The term numerical bounding is used to refer to the use of a quasi-stochastically consistent formula that prevents fractional collections exceeding unity for any moments. Through examples and analysis it is found that numerical bounding is typically important in mass moment prediction for time steps exceeding approximately 10 s. The Poisson-based numerical bounding scheme is shown to be simple in application and conceptualization; within a straightforward idealization it completely corrects overdepletion while even being immune to the rediagnosis error of the time-splitting method. The scheme’s range of applicability and utility are discussed.

Reversing the Proximity Effect in Civil Trials

2013 ◽  
Author(s):  
Kimberly Schweitzer ◽  
Scott Freng ◽  
Sean McCrea
Keyword(s):  
Proximity Effect ◽  
Civil Trials

Exact solution of a four-site crystal model for an intermediate-valence system

1986 ◽  
Vol 47 (6) ◽  
pp. 1029-1034 ◽  
Author(s):  
J.C. Parlebas ◽  
R.H. Victora ◽  
L.M. Falicov
Keyword(s):  
Exact Solution ◽  
Intermediate Valence ◽  
Crystal Model

Wave Propagation Characteristics on a Pipe-Type Cable in Particular Reference to the Proximity Effect

2013 ◽  
Vol 133 (12) ◽  
pp. 954-960 ◽  
Author(s):  
Akihiro Ametani ◽  
Kazuki Kawamura ◽  
Asha Shendge ◽  
Naoto Nagaoka ◽  
Yoshihiro Baba
Keyword(s):  
Wave Propagation ◽  
Proximity Effect ◽  
Propagation Characteristics
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