scholarly journals Numerical solution of a non-linear conservation law applicable to the interior dynamics of partially molten planets

2017 ◽  
Author(s):  
Dan Bower ◽  
Patrick Sanan ◽  
Aaron Wolf
Keyword(s):  
Numerical Solution ◽  
Conservation Law ◽  
Non Linear

Non-linear Convection at High Rayleigh Number

1970 ◽  
Vol 1 (8) ◽  
pp. 382-383 ◽  
Author(s):  
R. Van Der Borght
Keyword(s):  
Rayleigh Number ◽  
Numerical Solution ◽  
Asymptotic Method ◽  
Weak Coupling ◽  
Coupling Approximation ◽  
Non Linear ◽  
Weak Coupling Approximation ◽  
Basic Equations ◽  
Rayleigh Numbers ◽  
Steady Convection

The numerical solution of the basic equations for non-linear steady convection in the weak-coupling approximation is exceedingly difficult. In astronomical applications the Rayleigh number R will be of the order of1012, and we wish to report here some results obtained in the case of high Rayleigh numbers with the use of an asymptotic method.

Conservation Law of Energy Using Fractional Taylor Series

2016 ◽  
Vol 841 ◽  
pp. 105-109
Author(s):  
Ali Soroush ◽  
Farzam Farahmand
Keyword(s):  
Taylor Series ◽  
Conservation Law ◽  
Control Volume ◽  
Fractional Differentiation ◽  
Linear Flow ◽  
First Order ◽  
Taylor Series Approximation ◽  
Series Approximation ◽  
Non Linear ◽  
Flow Power

Customary conservation law of energy is commonly derived using first-order Taylor series, which is only valid for situation of linear changes in the flow of energy in control volume. It is shown that using high-order Taylor series will approximate non-linear changes in the flow of energy but in fact some error remains. We used fractional Taylor series which exactly represent non-linear flow of energy in control volume. By replacing the customary integer-order Taylor series approximation with the fractional-order Taylor series approximation, limitation of the linear flow of energy in the control volume and the restriction that the control volume must be infinitesimal is omitted. The innovation of this paper is we show that as long as the order of fractional differentiation is equal with flow power-law, the fractional conservation law of energy will be exact and it can be used for fluid in a porous medium.

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